LIBRARY OF CONGRESS. 

Chap.....\-D Copyright No. ._, 



Shelf. 



iKh3 



UNITED STATES OF AMERICA. 



AN ELEMENTARY TREATISE 



SURVEYING AND NAVIGATION 



ARTHUR G. ROBBINS, S.B. 

ASSISTANT PROFESSOR OF CIVIL ENGINEERING, MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY 








.-^x 



LEACH, SHEWELL, AND SANBORN 

BOSTON NEW YORK CHICAGO 



\ 



Copyright, 1896,' 

bt leach, shewell, and sanbokn. 






J. S. Cushing & Co. - Berwick & Smith 
Norwood Mass. U.S.A. 



/ ^0 Lf) 



PREFACE. 



This brief work ou Surveying and Navigation is intended for those 
students who desire to supplement the study of Trigonometry with a 
brief course on its applications to those subjects. 

No attempt has been made to treat the subjects fully. Special effort 
has, however, been made to have the work correct and accurate as far as 
it goes, and it is believed that the student who afterwards becomes a 
surveyor or navigator will have nothing to unlearn that he has learned 
from this work. 

The author wishes to acknowledge his obligations to Messrs. W. & 
L. E. Gurley, Troy, N.Y., for the use of plates from which were made 
the cuts of the instruments found throughout the work. 



ARTHUR G. ROBBINS. 



Massachusetts Institute of Technology, 
September, 1896. 



CONTENTS. 



PAGE 

The Chain and Tape ............ 1 

Measurement of Lines. 
Measurement of Angles. 
Passing Obstacles. 
Measurement of Areas. 



The Compass . ...» 5 

Measurement of Bearings. 

Declination of the Needle. 

Variation of the Needle. 

Declination Formulas. 

Kunning Old Lines. 

Local Attraction. 

Compass Surveys. 

To Straighten a Bent Pivot and Needle. 

Calculation of the Aeea 11 

Latitudes and Departures. 

Error of Closure. 

Balancing. 

Double Meridian Distance. 

Plotting 14 

The Protractor. 

Total Latitudes and Departures. 

Parting opf Land 16 

Running a Random Line ........... 10 

To straighten a Curved Boundary ......... 17 

The Transit 17 

The Plain Transit. 
The Engineer's Transit. 
Verniers. 

Measurement of Angles. 
Adjustments. 

Stadia Surveying ............. 23 



yi CONTENTS. 

PAGE 

PcBLic Land Surveys , ... 24 

The Solar Compass 26 

Determination of the Sun's Declination. 

The Level 27 

Profile Levelling. 
Cross-sectioning. 
Determination of Volumes. 
Adjustments. 

Contour Lines 33 

Practical Astronomy and Navigation 34 

Definitions. 
The Sextant. 
Lidex Error. 
Dip. 

Parallax. 
Refraction. 

Determination of Latitude 40 

Time . . • -^ 42 

Astronomical Time. 
Civil Time. 
Determination of Time. 

Establishment of a Meridian .44 

Navigation ......' . . 45 

The Mariner's Compass. 

Variation. 

Deviation. 

Leeway. 

The Log. 

The Knot. 

Length of an Arc of Longitude at Different Latitudes. 

Middle Latitude Sailing. 

The Chronometer . 49 

Determination of Longitude. 

Stadia Tables ,51 

Refraction Tables (To be used with Solar Compass) 59 



LA¥D SURVEYING. 



oXKc 



Surveying is the art of measuring and locating lines and angles on 
the earth's surface. 

Representation of these measurements on paper is called j^lotting. 

By applying the rules of geometry and trigonometry to these measure- 
ments, distances and quantities may be computed. 

In plane surveying all distances measured are horizontal distances. 

The instruments for measuring distances are as follows : 

Gunter's or surveyor's chain, made of iron or steel wire, contains 

100 links = 4 rods — 66 feet. 
1 link = Ty^o^o inches. 
10 sq. chains = 1 Acre. 

Engineer's chain, 100 ft. long, contains 100 links, each one foot long. 

The steel tape, which is made of a continuous ribbon of steel, generally 
in lengths of 50 or 100 ft., is graduated to feet, tenths and hundredths 
of a foot. 

The metallic tape, generally 50 ft. in length, is made of cloth, into 
which is woven a number of fine wires to jjr event stretching. It is gradu- 
ated to feet, tenths and half-tenths. 

Wooden rods, for measuring short distances, are graduated to read by 
a vernier to one-thousandth of a foot. 

The inch is not used in surveying field work. 

To measure a Straight Line with the Chain. 

The work is done by two persons irsing a chain and a set of eleven 
marking-pins. Sometimes two lining-poles are used to mark the ends of 
the line. 

The fore chainman takes ten of the eleven pins, and sets a pin at the 
end of each chain length, after the rear chainman has lined it in with the 
point to which the distance is to be measured. The rear chainman lines 

1 



2 LAND SURVEYING. 

in the pin to be set by the fore chainman, holds the rear end of the chain 
against the last pin set by the fore chainman until the next one is set, and 
takes up each pin after the next forward one is set. 

As soon as the fore chainman sets his last pin, he calls up and receives 
from the rear chainman his ten pins, taking care to count them to see that 
none are lost, records one tally, i.e. ten chains, and the work is continued 
till the end of the line is reached. The total distance is found by count- 
ing ten chains for each tally, one chain for each pin in the hand of the 
rear chainman, and the fractional part of a chain between the last pin set 
and the end of the line. The last pin should remain in the ground and 
not be counted as in the hand of the rear chainman. 

When a line is being measured down hill, the forward end of the 
chain should be held at the same level as the rear end, and the point 
transferred to the ground by a plumb line, or, less accurately, by a lining- 
jjole suspended between the thumb and finger. In all cases the chain 
must be held horizontal, whether the ground is level or not. The ten- 
dency, when learning to chain on a steep hill, is to hold the chain inclined 
at a less angle with the ground than the one between the horizontal and 
the surface of the gTound. 

Because of the large number of wearing surfaces, usually 600 in the 
surveyor's and engineer's chain, the length of every chain should be fre- 
quently compared with a standard. 

A new steel tape may be used as a standard when there is none other 
convenient. The comparison should be made on a smooth and level sur- 
face, such as a level sidewalk or long hallway. Provision is made for 
shortening the chain by turning a nut at the end of either one of the end 
links. 

If a line has been measured with a chain afterwards found to be too 
long or too short, the measured length should be corrected. In making 
this correction it should be remembered that the true length of a line, 
measured with a chain that is too long, is greater than the measured 
length, and vice versa. 

The degree of precision obtained ought to depend upon the character 
of the work, and not upon the character of the ground. With ordinary 
skill in chaining, the error in measuring a line along the highways of a 
town or village should not exceed one part in four or five thousand. If no 
more care be used in measuring a line through the tangled undergrowth 
of some forest lands, an error of one in three or four hundred might be 
expected. 

By applying the rules of geometry and trigonometry, it is practicable 
to measure angles, considerable areas, and distances to inaccessible objects 
by means of the chain. 



LAND SURVEYING. 3 

Given the line ab, to locate on the ground the line be jiei'pendicular to 
ab' at b. 

Select any convenient point as c, one chain length from b. With one 
end of the chain held at e, swing the other end to d, in the line ah, using 



a 


d 




b 




■\ 




/ 






^<?o 


4/ 






\ 








K 


e 



a lining-pole at d to aid in lining in from a if necessary. With one end 
of the chain still held at c, swing the other end to locate e in the line dc 
produced, be is the perpendicular required. 

Another Method. 

Measure cb = 40 ft. Hold one end of the chain at c, and the 80-ft. 
mark at b. Let a third person take the chain at the 50-ft. mark and pull 
both parts cd and hd taut, hd is the line required. 

Other methods equally feasible will suggest themselves to the student. 



40 




Fi&. 2. Fig. T,. 

Given a line ab on the ground, to locate a line starting at b and making 
an angle of 60° ivith this line. 

Prolong ab (Pig. 3) any convenient distance, as 100 ft., to c. Hold 
one end of the chain at b, and with the same length be locate a number 
of pins at d, f, etc., a foot or two apart and at a distance from c as nearly 
equal to be as can be estimated. Measure from c the distance ce = cb, 
lining in the point e between the tAvo adjacent pins (d and / in this case). 
be is the line required. 



LAND SURVP:YING. 



To lay out Any Given Angle, as 22° 30', with the Chain. 

By trigonometry, twice the sine of one-half of an angle is equal to its 
chord, the radius being unity. 

Given the line ab on the ground to locate a line cd, making the angle 
bed = 22° 30'. 

From c measure ce any convenient distance, as 100 ft. Twice the 
sine of one-half of 22° 30' = .3902. Multiply this by distance ce (100 ft.. 

a c e b 




Fig. 4. 

in this case). From e, with radius 39.02 ft., locate pins on arc^ one or 
two feet apart. From c, with radius 100 ft., locate the point h on the 
arc fg. cd passing through h is the line required. 

By the reverse of the above process, the angle between any two lines 
located on the ground may be measured. , : 

Figures 5, 6, 7, and 8 illustrate how lines may 
be prolonged through obstacles, and distances to 
inaccessible objects measured by application of 
these problems. 





% 



.^ 



60 




Ftg. 8. 

il) : ac: : ad: cd 



Areas of fields may be determined by measuring the length of lines 
dividing the field into triangles and computing the area of each triangle 



LAND SURVEYING. 5 

separately. In dividing the field into triangles, much more accurate 
results will be obtained if the lines to be measured are so chosen as to 
divide the field into as few and as nearly equilateral triangles as possible. 
The area of a narrow, irregular area such as that shown at abc (Fig. 9) 
can best be determined by measuring at regular intervals the lengths of 




Fig. 9. 



right-angle offsets to the lines ah and he and then computing the area by 
the following rule, known as the trapezoidal rule, 

in which ^i is the area, d the common distance between the offsets, h and 
li' the two end offsets, and 2 %hi twice the sum of all the offsets between 
the two end offsets. 

This rule assumes that the bounding line between any two adjacent 
offsets is a straight line. 



THE COMPASS. 

The compass, shown in Fig. 10, is an instrument for determining the 
angle that any line makes with the magnetic meridian. • 

It consists essentially of a circle, graduated to half-degrees, and num- 
bered from 0° to 90° each way from the north and south points. 

The sights are in the prolongation of the line IST. S. Balanced on a 
steel pivot at the centre of the circle, is a magnetic needle, which points 
in the direction of the magnetic meridian. The compass is connected with 
the tripod by a ball and socket joint, which allows the compass to be 
turned in any direction for the purpose of levelling. 

By setting the compass anywhere in a line and levelling it, by aid of 
the spirit bubbles shown on the compass box, then turning the sights till 
they point in the direction of the line, the angle which this line makes 



6 



LAND SURVEYING. 



with the magnetic meridian may be read, by observing the number of 
degrees between the N. or S. point of the compass box and the N. or S. 
end of the needle. This angle is called the bearing of the line. Bearings 

a 




are read from the N. or S. point of the compass box, 90° in each direction. 
The letters which mark the E. and W. points on the compass box are re- 
versed to aid in reading bearings, as is shown in Figs. 11, 12, 13, and 14. 

The bearing of the line ab, Fig. 11, is N. 20° W. 
The bearing of the line ab, Fig. 12, is N. 20° E. 
The bearing of the line ab, Fig. 13, is S. 20° E. 
The bearing of the line ab. Fig. 14, is S. 20° W. 




Fig. 12. 



LAND SURVEYING. 7 

' If the observer directs the N. point of the compass box in the direction 
of the line whose bearing is to be taken, and reads the N. end of the 
needle, the letters on the comj^ass box between which that end of the 
needle lies are the ones to be used in recording the bearing. A careful 
observance of this rule Avill prevent the student from reading bearings 
incorrectly when learning to use the compass. 

The magnetic needle does not point true north. Moreover, it does not 
ahvays point in the same direction. 

The angle which the magnetic needle makes with the true meridian 
is called the declination of the needle. Changes in the declination are 
called variations of the needle. The principal variations are, secular 
variation, daily variation, irregular variations. 

The secular variation is a slow and continuous change in the pointing 
of the needle, which may be compared to the movement of a pendulum 
requiring centuries to make a single oscillation. 

The daily variation is small, seldom exceeding six or eight minutes, 
and in ordinary surveying is neglected. 

Irregular variations may occur at any time. They follow no known 
law. 

In order to learn what the declination of the needle is at any time, 
the compass should be frequently set in a true meridian line ^ and the 
declination measured. 

The United States Coast and Geodetic Survey publishes from time to 
time, charts of the United States, on which are drawn lines connecting 
places at which the declination at any given time is the same. These 
lines are called isogonic lines. The line connecting places where there is 
no declination is called the agonic line. In 1890 this line passed a little 
to the east of Charleston, S.C., through North Carolina, Virginia, Ohio, 
and Michigan. At all places to the east of this line, the declination is 
west, and at all places to the west of this line the declination is east. 

At present the icest declinations are increasing about two to four min- 
utes a year, while the east declinations are decreasing ; or, in other words, 
the agonic line is moving west. 

From observations extending over a number of years it is possible to 
devise formulas by which the declination of the needle may be computed, 
with a considerable degree of precision, for some years in the future. 

The following are a number of such formulas taken from tables in the 
United States Coast and Geodetic Survey Eeport for 1886 : 

D = the declination, + when east, and — when west, m = time — 1850, 
expressed in years and fraction of a year. 

1 A method of establishing a true meridian line is explained in the chapter on 
Practical Astronomy. 



LAND SUEVEYING. 



Name of Station and State. 


Latitude. 


West 
Longitude. 


The Magnetic Declination expressed as a 
Function of Time. 


Montreal, Can 


45 30.5 


o ' 

73 34.6 


O O o 

Z)= + 11.88+4.17 sin (1.50m- 18.5) 


Eastport, Me 


44 54.4 


66 59.2 


Z)= + 15.14 + 3.90 sin (1.20m+ 31.7) 


Portland, Me 


43 38.8 


70 16.6 


Z>= +11.26 + 3. 16 sin (1.33m+ 5.8) 


Burlington, Vt 


44 28.5 


73 12.0 


Z>= + 10.81 + 3.65sin(l..S0m- 20.5) 
+ 0. 18 sin (7.00 u) + 1.32.0) 


Hanover, N.H 


43 42.3 


72 17.1 


D=+ 9.80 + 4.02sin (1.40m- 14.1) 


Eutland, Vt 


43 36.5 


72 55.5 


Z)= + 10. 03 + 3. 82 sin (1.50 m- 24.3) 


Portsmouth, N.H. . . . 


43 04.3 


70 42.5 


Z)=+10.71 + 3.36sin (1.44m- 7.4) 


Boston, Mass 


42 21.5 


71 03.9 


Z>=+ 9.48 + 2. 94 sin (1..30m+ 3.7) 


Hartford, Conn 


41 45.9 


72 40.4 


i)=+ .8.06+2.90 sin (1.25 m- 26.4) 


Albany, N.Y 


42 39.2 


73 45.8 


Z)=+ 8.17 + 3.02 sin (1.44m- 8.3) 


Harrisburg, Pa 


40 15.9 


76 52.9 


Z>=+ 2.93+2.98 sin (1.50 m+ 0.2) 


Baltimore, Md 


39 17.8 


76 37.0 


D=+ 3.20 + 2.57 sin (1.45 m- 21.2) 


Charleston, S.C. .... 


32 46.6 


79 55.8 


Z>=- 2.14 + 2.77 sin (1.40 m- 3.1) 


Key West, Fla 


24 33.5 


81 48.5 


D=- 3.70 + 3.16sin(1..35m- 35.1) 


New Orleans, La. . . . 


29 57.2 


90 03.9 


i>=- 5.01 + 2.57 sin (1.40m- 61.9) 


Cincinnati, 0. .... 


39 08.6 


84 25.3 


D=- 2.40+2.62 sin (1.42m- 39.8) 


Pittsburgh, Pa 


40 27.6 


80 00.8 


Z)=+ 1.85 + 2. 45 sin (1.45m- 28.4) 


Cleveland, 


41 30.3 


81 42.0 


Z>=+ 0.10 + 2.07 sin (1.40m- 6.2) 


Detroit, Mich 


42 20.0 


83 03.0 


D=- 0.97+2.21 sin (1.50m- 15.3) 


Buffalo, N.Y 


42 52.8 


78 53.5 


D=+ 3.66 + 3.47 sin (1.40m- 27.8) 


Sitka, Alaska . . . . 


57 02.9 


135 19.7 


Z)= -25.79 + 3.30 sin (1.30m-104.2) 


Port Townsend, Wash. T., 


48 07.0 


122 44.9 


i)= -18.84 + 3.00 sin (1.45 m-122.1) 


San Francisco, Cal. . . 


37 47.5 


122 27.3 


D= -13.94 + 2.65 sin (1.05m-135.5) 


Monterey, Cal 


36 36.1 


121 53.6 


Z)= -13.79+2.65 sin (l.lOm-156.4) 


San Diego, Cal 


32 42.1 


117 14.3 


Z)= -11.78 + 1.90 sin (1.15 m- 151.6) 



It is in relocating the lines of an old survey when the location of some 
of the corners has been lost that a knowledge of the changes in the 
declination of the needle is essential to the surveyor. For example, the 
declination of the needle at NeAv Haven, Conn., was 

5° 45' W. in 1761, 
5° 15' W. in 1780, 
4° 30' W. in 1819, 
5° 30' W. in 1828, 
6° 00' W. in 1838. 
Therefore, a line that had a bearing of N. 10° W. in 1761, had a bearing of 

N. 10° 30' W. in 1780, 
N. 11° 15' W. in 1819, 
N. 10° 15' W. in 1828, 
N. 9° 45' W. in 1838. 



LAND SURVEYING. 9 

At a place in Illinois the declination of the needle in 1821 was 8° 00' E., 
and in 1843 it was 7° 15' E. ; therefore a line that in 1821 had a bearing 
of S. 20° W., had, in 1843, a bearing of S. 20° 45' W. 

When the bearing of a line is being read, see that there is no iron or 
steel near, to turn the needle from its normal position. Some of the 
things most liable to cause this are, the chain or pins, hatchet, covered 
steel buttons on coat, steel-bowed spectacles, keys or knife in vest pocket. 
In fact, any bit of iron held near the needle may attract it appreciably. 

To measure the bearing of a line when local attraction cannot be 
avoided, as, for example, when the line is an iron fence or a railroad 
track, measure the bearing in the usual way, and also, from the same 
place, measure the bearing to a point where no local attraction exists. 
Next set the compass over this latter point and measure the bearing to 
the first point. This bearing being correct, the bearing of the fence or 
track may be computed. 

Example. — Compass at a (Eig. 15), bearing a6 == N. 20° E.; bearing 
ac = ]Sr. 85° E. ; therefore angle cah = 65°. 




,d 



c 

Pig. 15. 

Compass at c, bearing ca = N. 83° W., which is the true bearing of ca, 
provided there is no local attraction at c ; therefore the true bearing of ac 
is S. 83° E. ; and since angle hac — 65°, the bearing of ab is IST. 32° E. 
instead of N. 20° E. 

In order to prove that there is no local attraction at c, select a fourth 
point d and measure the bearing cd ; then at d measure the bearing dc. If 
one is exactly the reverse of the other, there is no local attraction at either 
c or d. 

To make a compass survey of a field, the bearing and reverse bearing 
of all the sides should be taken, and the length of all the sides measured. 



10 LAND SURVEYING. 

The following are samples of field notes of compass surveys : 





.2 

33 


Bearing. 


Eeverse 
Bearing. 


Distance. 


.2 


Bearing. 


Reverse 
Bearing. 


Distance. 










o 




o 


cliains 









o 


chains 


(1) 


1 


N. 


37 E. 


S. 


37 W. 


15.32 


1 


N. 


75} W. 


— 


5.22 


(3) 




2 


N. 


46.} W. 


S. 


46| E. 


4.53 


2 


S. 


77 W. 


— 


10.60 






3 


S. 


43} W. 


N. 


43} E. 


13.75 


3 


S. 


74| W. 


— 


4.57 






4 


S. 


26 E. 


N. 


26 W. 


5.00 


4 


N. 


86 E. 


— 


3.84 






5 


s. 


57 E. 


N. 


57 W. 


1.60 


5 

Q 


S. 
S. 

N. 


50 E. 
27} E. 
65} E. 


— 


4.00 
3.93 
7.90 




(2) 


1 


N. 


67 E. 


S. 


67 W. 


3.66 


7 









2 


*S. 


24} E. 


K 


24} W. 


0.95 


8 


N. 


23 E. 


— 


2.17 






3 


S. 


361 E. 


N. 


36} W. 


1.34 


9 


N. 


33 E. 


— 


1.00 






4 


s. 


53} E. 


N. 


53} W. 


2.00 


10 


N. 


46} E. 


— 


1.84 






5 


s. 


42J E. 


N. 


42 W. 


1.14 


11 


N. 


60} E. 


— 


1.40 






6 

7 


s. 
s. 


35} E. 

74} W. 


N. 


35} W. 

74} E. 


2.52 
3.20 
















1 


N. 


89} W. 


S. 89 E. 


4.74 


(4) 




8 


N. 


33 W. 


S. 


32| E. 


3.30 


2 


N. 


17} W. 


S. 17} E. 


12.50 






9 


N. 


50| W. 


S. 


50| E. 


1.77 


3 


S. 


73| E. 


N. 73i W. 


15.36 






10 


N. 


61| W. 


s. 


61} E. 


1.14 


4 


S. 


38} W. 


N. 38} E. 


9.87 






11 


N. 


47} W. 


s. 


47} E. 


1.53 















(2) Area 2 A. 29 Rds. (3) Area 7 A. 155 Rds. (4) Area 9 A. 127 Rds. Ans. 

If the readings of the two ends of the needle are not alike, the 
trouble is probably due to a bent needle or a bent pivot. If the needle is 
bent, the difference in the readings of the two ends will be the same, 
whatever the direction of the compass sights. If the difference in the 
readings of the two ends of the needle changes with the direction of the 
compass sights, the pivot is bent, and the needle may or may not be 
straight. 

To straighten the pivot, turn the compass box till the difference in the 
readings of the two ends of the needle is greatest, then bend the pivot at 
right angles to the direction of the needle. After the pivot is straight- 
ened, to straighten the needle, bend till the readings of the two ends are 
the same. 

To remagnetize the Needle. 

Remove the glass cover from the compass box and take the needle from 
the pivot. Place the needle on a fiat surface and draw an ordinary bar 
magnet from the centre of the needle to the end, repeating the operation 
a number of times. In returning the magnet from the end of the needle 
to the centre, it should be carried several inches away from the needle 



LAND SURVEYING. H 

and not' moved back close to its surface. Use the north pole of the 
magnet on the south end of the needle, and vice versa. 

Sometimes when the surface of the glass cover to the compass has 
been rubbed with a dry cloth, electricity will be developed and cause 
the needle to adhere to the glass when lowered to the pivot. A touch 
on the glass with the moistened finger will remove the difficulty and 
cause the needle to swing free. 

Always keep the needle raised from the pivot except tvhen taking a 
bearing, otherwise the sharp point of the pivot will become blunt, and the 
pointing of the needle much less precise. 

CALCULATION OF THE AREA. 

The latitude of a line, or its northing or southing, is the distance that 
one end of the line is north or south of the other end. 

It is equal to the length of the line, multiplied by the natural cosine 
of its bearing. 

The departure oi a line, or its easting or westing, is the distance that 
one end of the line is east or west of the other end. 

It equals the length of the line multiplied by the natural sine of its 
bearing. 

Obviously in a closed field the sum of the northings should equal the 
sum of the southings, and the sum of the eastings equal the sum of the 
westings. 

It is customary to consider north latitudes and east departures positive, 
and south latitudes and west departures negative. . 

Compute all the latitudes and departures. The difference between the- 
sum of the northings and the sum of the southings is the error in latitude. 
The difference between the sum of the eastings and the sum of the west- 
ings is the error in departure. The square root of the sum of the squares 
of these errors is the error of closure. This should not exceed a certain 
percentage of the perimeter, the amount depending upon the precision 
required for the work in hand. 

After determining the error in latitude and in departure, this error 
must be distributed among the several courses, in order that the latitudes 
and departures shall balance exactly. This may be done by the following 
rule : 

The sum of cdl the latitudes, or departures, is to any latitude, or de- 
parture, as the total error in latitude, or departure, is to the correction to be 
cq:)plied to that latitude, or departure. 

This simply distributes the error throughout the whole length of the 
survey. It would be more precise, if not scientific, to make the correc- 



12 LAND SURVEYING. 

tion to the lines or bearings tliat were incorrectly measured. The sur- 
veyor, from his knowledge of the survey, should distribute the error 
among those lines and bearings in which the natural difficulties to doing 
correct work were the greatest. 

The surveyor should be cautious about increasing the length of any 
of the lines, in order to balance the survey, since any chainman, however 
careful, is much more likely to make the measured length of a line too 
long rather than too short, because of the difficulty in drawing the chain 
perfectly straight and horizontal. 

After balaiacing the latitudes and departures, compute the double me- 
ridian distances. 

The double meridian distance of any line is equal to twice the distance 
of its centre from any chosen meridian. 

It will be found convenient to choose the meridian passing through 
the extreme east or west point of a survey, from which to compute the 
double meridian distances, in order that they may all have the same alge- 
braic sign. This point may generally be easily determined by a simple 
inspection of the field notes. Having selected this point, call the line 
starting from it the first course. 

Rule. — The double meridian distance of the first course is equal to its 
departure. 

The double meridian distance of the second course is equal to the double 
meridian distance of the first course, plus its dej^arture, p?«.s the departure 
of the second course. 

The double meridian distance of any course is equal to the double merid- 
ian distance of the preceding course, plus its departure, plus the departure of 
the course itself. 

The double meridian distance of the last course should equal its de- 
parture, which checks the computation. 

Multiply the double meridian distance of each course by its corrected 
northing or southing, observing that the product of a northing multiplied 
by a positive double meridian distance gives a positive result, and that 
the product of a southing multiplied by a positive double meridian dis- 
tance gives a negative result. 

The algebraic sum of all these products is twice the area of the field. 



LAND SURVEYING. 



13 



' The following is the computation of the area from the lield notes given 
in Example 1, page 10 : 





Bearing. 


(5 


Latitude. 


Dciiartm-e. 


Ralanood. 


D.M.D. 


+Areas. 


—A re IS. 


N+ ; s- 


K + 


w- 


T .-t , Depart- 


1 


o 

N.37 E. 


15.32 


12.24 




9.23 




1 

-t- 12.23 1 + 9.22 


16.28 


199.104 




2 


N. 46.V W. 


4.53 


3.11 


— 


— 


3.28 


+ 3.11 -3.28 


22.22 


69.104 


— 


3 


S. 43^ W. 


13.75 


— 


9.97 


— 


9.46 


- 9.98 


-9.47 


9.47 


— 


94.511 


4 


S. 26 E. 


5.00 


— 


4.49 


2.19 


— 


- 4.49 


+ 2.19 


2.19 


— 


9.833 


5 


S. 57 E. 


1.60 


— 0.87 


1.34 


— 


- 0.87 


+ 1.34 


5.72 


— 


4.976 



error in soutliing, 0.02 



15.35 15.33 12.76 12.74 
15.33 12.74 

0.02, error in westing. 



268.208 109.320 
109.320 



2)158.888 




79.444 sq. chains 
■ acres 151.1 rods. 



Fig. 16. 



ABODE (Fig. 16) shows a plot of the field, the area of which is com- 
pnted above. MN is a meridian through A, the most westerly point. 
(Sta. 4 in field notes.) 



Course. 


Latitude. 


D.M.D. 


H- Areas. 


— Areas. 


AB 


AI 


2ab 




2ABI 


BC 


IN 


2cd 




2IBCN 


CD 


No 


2ef 


2 NCBo 




BE 


oM 


2gh 


2 oDEM 




EA 


MA 


2ik 




2 AME 



14 



LAND SURVEYING. 



It is clear from Fig. 16 that each area given in the last table is equal 
to the product of the latitude of the course forming one of its limits, 
into the double meridian distance of that course. It is also evident that 
the difference between the + areas and the — areas is equal to twice the 
area ABODE. 

Plotting. 

A survey may be plotted by laying off the angles with a protractor, 
and the distances, to the desired scale, with any convenient form of 
scale ; or the corners of the field may be plotted by means of total 
latitudes and departures. 

The protractor, one form of which is shown in Fig. 17, is generally 
made of a circular or semi-circular piece of brass or German silver, gradu- 
ated to degrees on its circumference. 




To lay off an angle with the protractor, place the centre mark over the 
vertex of the angle, Avith the diameter through the 0° mark along the 
given line. Mark off the number of degrees contained in the angle, and 
connect this point with the vertex. 

In every case the line from which the angle is laid off should extend 
beyond the circumference of the protractor in both directions, to insure an 
accurate result. 



To plot by Total Latitudes and Departures. 

The total latitude, or departure, of any station is equal to the algebraic 
sum of all the latitudes, or departures, of all the preceding courses. 



LAND SURVEYING. 



15 



, The following are the total latitudes and departures of the survey 
computed on page 13 : 



Station. 


Total Latitude. 


Total Departure. 


A 








B 


-4.49 


+ 2.19 


C 


- 5. .36 


+ 3.5.3 


D 


+ 6.88 


+ 12.75 


E 


+ 9.98 


+ 9.47 


A 









Draw a meridian line, and locate at any convenient point along this 
line the point having for its total latitude and departure. Lay off 
above this point the greatest positive total latitude, and below this point 
the greatest negative total latitude. At each of these latter points erect 
perpendiculars, and give each a length equal to the greatest total depart- 
ure. Draw a line connecting the ends of these perpendiculars, and a 
rectangle will be formed, inside of which the plot is to be located. From 
these lines locate each station of the survey, using the total latitudes as 
ordinates and the total departures as abscissas. Draw lines connecting 
each adjacent station, and the survey is plotted. To check the work, scale 
each line and see if it agrees with the measured length. {ABODE, Fig. 
16, is plotted by total latitudes and departures.) 

Every plot should show, in addition to the outline of the field, the 
length and bearing of each line, the date of the survey, the declination of 
the magnetic needle at the time the survey was made, the scale of the 
plot, the location of the field, and the name of the surveyor. 

When it is not practicable to measure one line of a survey, the missing 
data may be supplied by the computation, as in that case the difference 
between the northings and the southings is the latitude of the missing 
line, and the difference between the eastings and westings is its departure. 
Divide the departure by the latitude, and the quotient is the natural 
tangent of the missing course. If practicable, all of the courses and dis- 
tances should be measured, as whenever one is omitted the whole error of 
closure is, of necessity, thrown into the missing line, .and the surveyor 
has no knowledge of the magnitude of the error. 



16 



LAND SURVEYING. 



PARTING OFF LAND. 



Suppose that in the field plotted on page 13, it is required to locate 
from a point F (Fig. 18) on the line CD, eight chains from C, a line FX 




Fig. 18. 



such that the area ABCFX shall be four acres. It is evident that the 
point X will fall somewhere on the line AE. 

Draw AF. -Compute the area ABGF. 

The latitude, departure, and bearing of AF may be determined as in 
the case of the missing bearing and distance of any line. Subtract the 
area ABCF from four acres, and there remains the area of the triangle 
AFX. Compute the length of PX, perpendicular to AF (^-AFxFX 
= area AFX). With the angle XAF and the length PX given, AX 
may be computed, and X located by measuring this distance from A. 



RUNNING A RANDOM LINE. 

If for any reason, in running from one station of a survey to another, 
the two points are not intervisible, and there is no dividing fence, a ran- 
dom line is first run in as nearly the right direction as can be jvidged, till 
a point is reached, at which an offset at right angles to the random line 




will pass through' the other end of the true line. Stakes should be set 
at regular intervals, say one chain apart, along the random line, from 
which points on the true line may be located as shown in Fig. 19. 

ab = 8.75 eh., be — .35 ch. ; then to locate a point on the line ac oppo- 

Q 

site the stake at 8 the offset is t; — - x .35^.32 ch. The offset from the 

o. ( O 

stake at 7 is .28 ch., and so on. 



LAND SURVEYING. 17 

, If the line divides two wood lots, one of which is to be cut, the line 
is marked by blazing adjacent trees on the side nearest the line, as 
shown in Fig. 20. 



To replace a broken boundary line by a single straight line, without 
changing the area of the adjoining fields. 

Let ab (Fig. 21) be a curved or broken line separating the two fields. 
It is required to locate a line, extending from a to the line cd, such that 
the area on either side of this line will be the same as that on either side 
of the line ab. 



Fig. 21. 

From a run a random line ac. From this random line measure offsets 
to the curved line ab and compute the area on either side of ac between 
it and ab. Divide the difference of these areas by half of ac and measure 
this distance, ce in the figure, from c, perpendicular to ac ; through e, par- 
allel to ac, lay off ef to cd. af is the line required. 

THE TRANSIT. 

The transit, two forms of Avhich are shown in Fig. 22, is an instru- 
ment for measuring angles independent of the magnetic needle and with 
much greater accuracy. 

The telescope differs from an ordinary telescope in that there is placed 
between the object-glass and the eye-piece, a ring carrying two fine spider 
threads called " cross-hairs." These are used to define the line of sight 
through the telescope, in the same way as do the compass sights on the 
compass. 

The telescope carries with itself an index, which moves about a gradu- 
ated circle and marks the number of degrees and minutes contained bc- 
tAveen two successive sightings of the telescope. 

The circle is graduated to half-degrees, or sometimes into twenty- 
minute spaces. The index carries a vernier, which is a device for read- 
ing accurately fractional parts of a degree. The single-minute vernier is 
generally made as follows : 



18 



LAND SURVEYING. 



A space on the vernier, equal to twenty-nine lialf-degrees on the circle, 
is divided into thirty equal parts. This makes each division on the 
vernier one-thirtieth of half a degree shorter than a half-degree, or one 
minute shorter. 




If the zero of the vernier is set opposite the zero of the circle, the 
lirst division of the vernier falls one minute short of the first division of 
the circle, the fifth division five minutes short, and so on, till the thirtieth 



LAND SURVEYING. 19 

division on the vernier is reached. This division falls thirty minutes 
short of the thirtieth division on the circle, and coincides with the twenty- 
ninth half-degree division. 




Fig 22 — Engineer's Transit. 



By moving the vernier till the first division from the zero coincides 
with the first division of the circle, the telescope is turned through an 
angle of one minute. By turning the vernier till its fifth division coin- 



20 LAND SURVEYING, 

cides with the lifth division of the circle, the telescope is moved through 
an angle of five minutes, and so on. 

To read the angle for any setting of the vernier, count, on the circle, 
the number of degrees and half-degrees between the zero of the circle 
and the zero of the vernier ; then look along the vernier, till a division is 
found which coincides exactly with one of the divisions on the circle ; the 
number of this division on the vernier is the number of minutes to be" 
added to the degrees and half-degree, if any, to give the true reading. 

The following is a general rule for determining the smallest reading of 
any vernier : 

Divide the value of the smallest division on the circle by the number of 
divisions on the vernier. 

For example, thirty divisions on vernier and circle divided to thirty- 
minute spaces, 

— = 1'. (1) 

Forty divisions on vernier ; circle divided to tAventy-minute spaces, 

20' 

— = 30". (2) 
40 ^ ^ 

Sixty divisions on vernier ; circle divided to twenty -minute spaces, 

90' 

^ = 20". (3) 

Sixty divisions on vernier ; circle divided to ten-minute spaces, 

10' 

1=10". (4) 

Ten divisions on vernier ; rod divided to hundredths of a foot, 

•-^^^ = .001 ft. (5) 

10 ^ ^ 

Ten divisions on vernier ; scale divided to tenths of an inch, 

^=•01 in. (6) 

Twenty-five divisions on vernier ; scale divided to .05 in., 

■■^^^ = .002 in. (7) 

25 



LAND SURVEYING. 2^ 

(1),- (2), and (3) are different forms of transit verniers, (4) is a sextant 
vernier, (5) is the form of vernier used on Boston and New York levelling- 
rods, and (6) and (7) are two forms of vernier used on mercurial 
barometers. 

To measure an Angle with the Transit. 

Bring the plumb-bob, hung from the transit, exactly over the point at 
which the angle is to be measured. Place the plate bubbles parallel to 
opposite levelling-screws and level, by grasping opposite levelling-screws 
betAveen the thumb and forefinger of each hand and turning both thumbs 
in or out, as is necessary. (The bubble will move in the same direction as 
the left thumb in turning the levelling-screws.) 

Set the zero of the vernier opposite the zero of the circle ; focus the 
eye-piece on the cross-hairs, and the object-glass on the object to be 
sighted; tighten the lower clamp and bring the image of the object, 
defining one line of angle, in exact coincidence with the vertical cross- 
hair, by means of the lower tangent screws. Loosen the upper clamp ; 
sight the telescope to the object defining the second line of the angle; 
tighten the upper clamp and bring the image of the object in exact 
coincidence with the vertical cross-hair, by means of the upper tangent 
screw. Kead on the circle the number of degrees passed over by the 
index and on the vernier opposite the line in coincidence with a line on 
the circle the number of minutes to be added to the circle reading, to give 
the correct reading. 

Generally there are four verniers on a transit, one on each side of the 
zero of each of the two opposite indices. The student, when learning to 
use the transit, should read that vernier which lies in the same direction 
from the zero set, as that in which the zero of the vernier was turned in 
sighting the second object. 

In making a survey of a closed field with the transit, either the 
interior angles or the deflection angles may be measured. If the interior 
angles are measured, their sum should equal twice as many right angles 
as the field has sides, less four right angles. If the deflection angles are 
measured, the sum of all the right deflections should differ from the sum 
of all the left deflections by 360°. 

In either case the bearing of each line should be measured with the 
needle. The bearing of each line should also be computed by assuming 
the computed bearing of one line (generally the first) equal to its observed 
bearing, and then computing the bearing of each succeeding line from the 
deflection angle. If the computed bearing does not agree approximately 
with the observed bearing, both should be remeasured to discover if any 
mistake has been made. 



22 



LAND SURVEYING. 



In Fig. 23, a, 6, d, e, and / are right deflection angles ; c and g are left 
deflection angles. 

Areas may be determined from the computed bearings and the dis- 
tances in the same way as with compass surveys. 




Fig. '-'o. 



Adjustments of the Transit. 

First. To make the plate bubbles parallel to the horizontal circle. 
Level the bubbles in the ordinary way ; then revolve the instrument half- 
way round. If either bubble runs toward one end of its tube, turn the 
adjusting screw at the end of the bubble tube till the bubble moves half- 
way back to the centre. Repeat the operation till the bubble will remain 
in the centre during a whole revolution of the instrument. 

Second. To adjust the line of collimation. This consists in placing 
the vertical cross-hair in the optical axis of the telescope, so that a 
straight line may be prolonged by revolving the telescope on its hori- 
zontal axis. 

Sight the telescope on some point A 200 to 300 ft. away, and 
clamp the spindle; then revolve the telescope on its horizontal axis 
and locate a point B in the line of sight. Loosen the clamp and 
revolve the instrument on its vertical axis till A is sighted. Clamp 
the spindle and again revolve the telescope on its horizontal axis, 
and note whether the line of sight strikes the point B. If it strikes to 
one side, call this point C. To adjust, move the cross-hair ring till the 
line of sight strikes a point one-fourth of the distance from C toward B. 
Test the correctness of the adjustment by repeating. To move the cross- 
hair ring, loosen the capstan-headed screw on one side of the telescope 
tube and tighten the opposite one. Unless the telescope is " inverting," 
the cross-hair ring must be moved in the direction opposite to what 
appears to be correct. 

Third. To make the axis of the telescope horizontal, so that the line 
of sight will move in a vertical line. Set up the instrument near some 
high object, as a steeple ; sight the high point and clamp the spindle. 
Depress the telescope and locate a point, in the line of sight, nearly on a 



LAND SURVEYING. 23 

level with the telescope. Loosen the clamp ; revolve the instrument on 
its vertical axis and the telescope on its horizontal axis ; sight the Ioav 
point and clamp the spindle. Raise the telescope and note if the line of 
sight strikes the high point. If the line of sight is to one side, the 
standard on the opposite side is too high, and the axis of the telescope 
must be moved till the line of sight moves half-way back toward the high 
point first sighted. To move the axis of the telescope, turn the screw 
under the bearing at one end of the axis. Repeat the operation in order 
to see if the adjustment has been correctly made. 



STADIA SURVEYING. 

In addition to the centre horizontal cross-hair in the telescope, most 
modern engineers' transits have two other horizontal hairs, situated at 
equal distances above and below the centre one. These are called stadia 
hairs. 

They are placed at such a distance apart that the distance intercepted 
by them, on a rod held vertical, is one-hundredth of the distance to the 
rod, from a point in front of the object-glass equal to the focal length of 
the telescope ; or, expressed in formula, 

d = 100s+f+c. 

d = distance from the centre of the instrument. 

s = space on rod intercepted by stadia hairs. 

/= focal length of telescope, i.e. distance from the object-glass to the 
cross-hairs = from three-fourths to one foot in most transits. 

c = distance from the object-glass to the centre of the instrument (= one- 
half foot, about). 

If the rod is not at the same level as the transit, and is still held ver- 
tical, the horizontal distance and the difference in elevation may be found 
by the following formulas : 

Horizontal distance = scos"'t' + (c +/) cos y. 
Difference in elevation = s ^ sin 2v-\-(f+c) sin v. 
V = vertical angle (either elevation or depression). 

The difference in elevation given, is the vertical distance between the 
centre of the telescope and the point on the rod intercepted by the centre 
horizontal cross-hair. By sighting at a point on the rod equal to the 
height of the telescope, the difference in elevation between the surface 
of the ground at the instrument and at the rod is given. 



24 LAND SURVEYING. 

The tables computed by Mr. Arthur Winslow of the State Geological 
Survey of Pennsylvania, give values of scos^ v and i s sin 2 v for angles 
from 0° to 30° with s = 1. By use of these tables the reduction of 
stadia notes is made quite easy. 

Example in Use of Tables. 

Vertical angle = + 4° 28' (c +/) = !. 25 = c in table. 
Rod reading = 4.42 ft. 

Difference in elevation = 4.42 x 7.76 + .10 = 34.40 ft. 
Horizontal distance = 4.42 x 99.39 + 1.25 = 440.55 ft. 

The stadia furnishes a very rapid method of measuring distances when 
an error of one part in four or five hundred is admissible. 

PUBLIC LAND SURVEYS. 

The larger part of the land north of the Ohio River and west of the 
Mississippi is laid out according to law, by lines running north and south, 
and east and west. 

The method is as follows : 

A principal meridian is run due north. At intervals of twenty-four 
miles north of latitude 35° and at intervals of thirty miles south of this 
latitude, standard parallels are run due east and west. At intervals of 
forty-two miles, lines are run north and south, connecting standard 
parallels. 

This divides the surface into figures shaped as shown in Fig. 24. 



Fig. 24. 



Each of these approximately rectangular plots, or checks, as they are 
called, is divided into townships which are six miles square, "as nearly 
as may be." The bounding lines of the townships are run as follows : 



LAND SURVEYING. 



25 



' Starting at «, six miles from o (Fig. 25), run a line due north six miles to 
!), establishing section and quarter-section corners every one-half mile. From 
b run a random line toward c. c has been already established on the prin- 







1 1 1 "^ 
\ \ \ k 


/ 





— - 


----1-4-^ 


(' 


f/ 


c 


y w 


X 


.___ J ! _- \ - ^ 


c 






1 1 1 

1 ' « 






eipal meridian. In all probability the random line will not strike c ex- 
actly, but somewhere north or south of it. In this case the true line must 
be run from c to & by correcting each one-half-mile point established on 
the random line. In the same way run from h to d due north ; then a 
random line from d toward e ; then correcting this from e to d. 

AVhen the line between the sixth and seventh township lines is run, a 
random line is run both east and west from ?7; toward x and y, and so on 
till the whole check is divided into townships. 

Each township is subdivided into sections numbered as shown in 
Fig. 26. 

Each section is divided into quarter-sections called respectively the 
iST.E., K.W., S.E., and S.W. quarter-section. 



6 


5 


4 


3 


3 


1 


7 


S 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


23 


23 


24 


30 


29 


38 


27 


26 


25 


31 


32 


33 


34 


35 


36 



Fig. 26. 



26 



LAND SURVEYING. 



The Hues of townships running east and west from a principal merid- 
ian are called toionships. The lines of townships running north and south 
of a standard parallel are called ranges; thus 



N.E. 1 sec. 18 T. 3 N. E. 5 W. 



is the northeast quarter of section eighteen, in the township in the third 
line of townships north of a standard parallel, and the fifth range west 
of a principal meridian. 

Burt's solar compass, shown in Fig. 27, is the instrument commonly 
used in running lines on government surveys. 




Fig. 2T. 

This instrument is devised to determine a true north and south line 
by means of an observation on the sun. 

At a is the latitude arc, on which is laid off the latitude of the place ; 
h is the declination arc, on which is laid off the declination of the sun 
(corrected for refraction) at the time when the observation is made. 

The compass is then turned about on its vertical axis, and at the same 
time the arm h about the polar axis p, till an image of the sun, formed 
by a lens at g, falls between the four lines on the silvered plate at u. 
The polar axis p is then parallel to the axis of the earth, and the compass 



LAND SURVEYmG. 27 

sights are in a true north and south line. Do not confuse the false image, 
sometimes reflected from the arm h, with the true image formed by the 
sunlight passing directly from the lens to the silvered plate. The false 
image is much less bright, and its outline is less clearly defined than that 
of the real image. 

To determine the Declination of the Sun at Any Time and Place. 

The Nautical Almanac gives in advance the declination of the sun for 
apparent and for mean noon at Greenwich for each day in the year. It 
also gives the change in declination for one hour. By multiplying this 
change by the number of hours after noon and adding the result to or 
subtracting it from the declination at noon, according as the declination is 
increasing or decreasing, the declination for any time may be determined. 

Having the declination at Greenwich, the time when the declination is 
the same at any other place is known, if the difference between local and 
Greenwich time is known ; or, in other words, if the longitude is known. 
For example, the declination in New England at 7 a.m. is the same as it 
is at mean noon, of the same day, at Greenwich, since the longitude of 
New England is five hours west of Greenwich. The declination in the 
Middle States at 10 a.m. is the same as that at Greenwich at 4 p.m. of the 
same day, since their longitude is six hours west of Greenwich. 

When the declination is north, the correction for refraction is to be 
added to the declination obtained from the Nautical Almanac, and when it 
is south, the refraction correction is to be subtracted, to give the angle to 
be laid off on the declination arc of the solar compass.^ 

Tables giving the correction for refraction for different hours of the 
day, for different latitudes, and for different declinations, have been com-^ 
puted. Those published by Messrs. W. & L. E. Gurley in their manual 
were prepared especially for use in connection with the solar compass. 



THE LEVEL. 

This instrument, shoAvn in Fig. 28, is used to determine the difference 
in elevation of different stations. 

It consists essentially of a telescope, attached to the under side of 
which is a delicate spirit level. The whole, mounted on a tripod, may be 
freely revolved about a vertical axis. 

The instrument is levelled by bringing the bubble into the centre of 
the tube when successively parallel to one set of opposite levelling-screws 

^ This is true for places north of the equator ; for places south of the equator the 
words north and south should be interchanged. 



28 



LA^D SURVEYING. 



and then parallel to the other set. The levelling-rod, two forms of 
which are shown in Fig. 29, is used to measure the distance down from the 
level line of sight to any point the elevation of which is desired. 




Fig. '28. 



To determine the difference in elevation between two points, the in- 
strument is set between them and levelled, and the distance of each point 
below the line of sight determined. The difference between these dis- 
tances is the difference in elevation. 

To determine each of these- distances, the rcjdman holds the rod on one 
of the points, and moves the target up, or down, as directed by the leveller, 
till the horizontal line on the target exactly coincides with the line defined 
by the horizontal cross-hair in the telescope. He then records the rod 
reading. The rod is then taken to the second point, the telescope is re- 
volved on its vertical axis till the rod is sighted, the target is again set, 
and the rod reading taken. The leveller should see to it that the bubble 
is in the centre of the tulje at the instant of sighting, as the fact that the 
instrument has been once levelled will .not insure its remaining so. If, 
for any reason, the difference in elevation of the two points cannot be 
determined by a single setting of the instrument, determine successively 



LAND SURVEYING. 



29 



the difference in elevation of a nnmber of intermediate points, as shown 
in Eig. 30. 

The ala:ebraic sum of tliese differences is the difference in elevation. 



^ 



^ 






Profile Levelling. 

In order to make a profile of the surface of the 
ground along any given line, elevations are taken 
at equal (generally 100 ft.) intervals. The eleva- 
tions are measured above an assumed horizontal 
plane called the datum plane. This plane should be 
below the lowest point on the surface, to avoid nega- 
tive readings. 

A bench-mark (B. M.) is some permanent mark, 
the elevation of which has been determined and 
recorded together with a description, by which it 
may be found at any time. 

A height of instrument (H. of I.) is the elevation 
of the line of sight through the level above the 
datum plane. It is found by adding a back-sight to 
the elevation of a bench-mark or turning-point. A 
back-sight or -f- sight (B. S.) is a rod reading taken 
on a bench-mark or turning-point, and is used to 
determine the height of instrument. A fore-sight, 
or —sight (F. S.) is a rod reading taken on a point, 
the elevation of which is to be determined. This 
elevation is determined by subtracting the fore-sight 
from the height of instrument. 

A turning-point (T. P.) is a point on which both 
a back-sight and a fore-sight are taken. It is used 
to determine a new height of instrument. A turn- 
ing-point should be solid, and not one whose eleva- 
tion can be changed by pressure of the rod while 
the sights are being taken. 

To determine the elevations for a profile, stakes 
are first set along the line at intervals of 100 ft., 
and numbered 0, 1, 2, 3, etc., to the end of the j-ig. 29. 

line. 

The level is set near the station, and a back-sight is taken on the 
nearest bench-mark. Fore-sights are then taken on as many stations as 
can be reached from that setting of the level.' A turning-point is then 
chosen and a fore-sight taken on it. The level is then carried forward 
along the line and set up again and levelled. A back-sight is then taken 



30 



LAND SURVEYING. 



on the turniiig-poiiit, a new height of instrument deteiniined, and tlien 
fore-sights are taken on as many more stations as may be sighted from 
this new position of the level. In this way the work is carried on till the 
end of the line is reached. 




Fig. 30. 



The following is a set of profile level notes : 



station. 


B. S. 


H. of I. 


F. S. 


Elevation. 


B. M. 


7.200 


114.900 




107.694 









4.2 


110.7 


1 






5.1 


109.8 


2 






6.3 


108.6 


8 






4.r 


110.0 


T. P. 


5.182 


116.210 


3.872 


111.028 


4 






4.2 


112.0 


+ 40 






5.7 


110.5 


5 






2.7 


113.5 


6 






1.8 


114.4 


B. M. 






0.987 


115.223 



Figure 31 shows a profile plotted from the above notes. 
In this way profiles for railroad lines, water-pipe lines, highways, etc., 
are determined. 




3 

Pig. .31. 



4 + 40 5 



Wlienever excavation or filling is to be made over considerable areas,, 
as in excavating for reservoirs, filling marshes, etc., the amount of material 



LAXD SURVEYING. 



31 



to 'be moved may be determined by what is known as cross-sectioning the 
area. This is done by running transit lines at right angles to each other 
to divide the field into squares, and then determining the elevation at 
each intersection. From these elevations, and the area of the total number 
of squares, the amount of material above any plane may be computed by 
multiplying the area of each square by the mean of the heights of the 
four corners above the given plane. 



Let Fig. 32 represent a piece of land 100 ft. square, and the figures at 
each corner the elevations in feet ; then the cubic contents above a two- 
foot horizontal plane are — i — — — ^^ x 100 x 100 cu. yds. 
^ 4 X 27 ^ 

When there are a large number of squares, take the sum of the heights 
at all of the corners common to one square, plus twice the sum of the 
heights at all the corners common to two squares, plus three times the 
sum of the heights at all the corners common to three squares, plus four 
times the sum of the heights at all of the corners common to four squares; 





9 




8 







11 



3 

Fi.:. 33. 



divide the result by 4, multiply by the area of one square, and the result 
is the contents in cubic feet. Divide by 27 to reduce to cubic yards. 



32 LAND SURVEYING. 

Figure 33 re]presents a plot of land, the area of each square being 
100 ft. The figures at the corners are the elevations in feet. The vol- 
ume in cubic yards above a zero plane is : 

100xl00x^^ + ^+'+^+^ + -^^ + ^ + ^ + ^^+^^^+^^^=13703.7c.u.yds. 

4 X 27 

Adjustments of the Wye Level. 

First. The adjustment of the line of collimation. This consists in 
placing the intersection of the cross-hairs in the centre of the wye-rings. 
Loosen the clips that hold the telescope in the wyes. Sight the vertical 
cross-hair on some well-defined line 200 to 300 ft. away, and clamp 
the spindle. Turn the telescope half-way round in the wyes and note 
if the vertical cross-hair still coincides with the line first sighted. If 
it does not, bring the line of sight half-way back to this position by 
moving the cro'ss-hair ring as is done in adjusting the line of collimation 
in the transit. Adjust the horizontal cross-hair in the same way. When 
this adjustment is made, the intersection of the cross-hairs will remain on 
a point while the telescope is turned through n whole revolution in the 
wyes. 

Second. To make the bubble parallel to the line of sight. Bring the 
telescope parallel to two opposite levelling-screws and clamp the spindle. 
Bring the bubble into the centre of the tube by turning the levelling-screws, 
and then turn the telescope a few degrees in the wyes. Should the bub- 
ble move toward one end of the tube, it would show that a vertical plane 
through the axis of the telescope is not parallel to a vertical plane through 
the bubble tube. To correct this, move the bubble to the centre by turn- 
ing the two capstan-headed screws, giving a horizontal motion to one end 
of the bubble tube. 

Repeat if necessary, till the bubble will remain in the centre when the 
telescope is revolved in the wyes five or ten degrees either side of its 
normal position. Next reverse the telescope in the wyes, and note if the 
bubble moves towards one end of the tube. If it does, that end. is too 
high and must be lowered, or the opposite one raised, till the bubble takes 
a position half-way back toward the centre. This is done by turning the 
capstan-headed screws, giving a vertical motion to one end of the bubble 
tube. 

Third. To make the line of sight and the bubble at right angles to the 
vertical axis of the instrument. Fasten the clips holding the telescope 
in the wyes ; bring the bubble parallel to two opposite levelling-screws, and 
level ; revolve the instrument 180° on the vertical axis, and if the bubble 
moves toward one end of the tube, turn the capstan-screws at the end of 



LAXD SURVEYTXf;. 33 

tlie wyes till the bubble moves half-way back to the centre. Kelevel, and 
repeat the operation to test the adjustment. 

Contour lines are lines connecting points of equal (devation. 

A shore line of a pond represents a contour line. If the water in the 
pond should rise a given amount, as 10 ft., then that shore line would 
represent another contour line 10 ft. higher than the first. 

Contour lines furnish an easy and accurate way of representing on a 
map the relief of a region. 

Such maps are valuable for man}' engineering purposes, among which 
are, location of routes for roads, railroads, water-pipes, aqueducts, luili- 
tary movements, etc. 

Contour maps on a small scale are often made by interpolating the 
contour lines on a plot that has been cross-sectioned as shown in Fig. 34. 




PRACTICAL ASTRONOMY AND NAVIGATION. 

-^wjisejoo 

The navigator, and sometimes the surveyor, needs a knowledge of as- 
tronomy sufficient to enable him to determine his position (latitude and 
longitude) on the surface of the earth. In the case of the navigator the 
error should not exceed the distance from the ship's deck to the horizon 
line, a distance seldom exceeding five or six miles. 

The zenith is the point vertically above the observer, and is 180° from 
the direction taken by a plumb-bob. 

The nadir is the point vertically underneath the observer, and lies in 
the direction taken by a plumb-bob at rest. The-zenith and nadir change 
with every change in the observer's position. 

The plane of the horizon is a plane passing through the observer's 
position, and everywhere at right angles to the direction of the zenith. 
Planes at right angles to the plane of the horizon, and containing the line 
connecting the zenith and nadir, are called planes of altitude. 

The two celestial poles are the points where the axis about which the 
earth revolves, would, if produced, pierce the heavens. The one which 
is above the horizon in the northern hemisphere is called the north celestial 
jjole ; the other is called the south celestial p)ole. 

The two poles are independent of the observer's position. 

The circle in which a plane passing through the earth's centre, and at 
right angles to the line connecting north and south poles, cuts the surface 
of the earth, is called the terrestrial equator. If this plane be extended, 
the circle in which it cuts the heavens is called the celestial equator. 

Hour circles are circles containing the line connecting the poles and 
at right angles to the plane of the equator. 

The plane containing the hour circle passing through the zenith inter- 
sects the horizon in its north and south points. 

The cdtitude of a point is its angular distance above the horizon, meas- 
ured in a vertical circle passing through the point. The zenith distance 
is the complement of the altitude. 

The azimuth of a point is the angle at the zenith between the meridian 
and a vertical circle passing through the point. Azimuth is generally 

34 



ASTRONOi\IY AND NAVIGATION. 



35 



mfeasured from the south through the Avest, north, and east, from 0° to 
360°. 

The declination of a point is its angular distance from the equator 
measured on an hour circle passing through the point. Declination is 
considered positive when the point is north of the equator, and negative 
when south of it. Polar distance is the complement of the declination. 

The hour angle of a point is the angle at the pole between the meridian 
and the hour circle passing through the point, or it may be defined as the 
arc of the equator intercepted by these two circles. Hour angles are usually 
measured from the south point of the equator from 0° to 360°, or from 
zero hours to twenty-four hours in the direction of the motion of the 
hands of a watch. 

The latitude of a point on the earth's surface is equal to the altitude 
of the pole. The observer's latitude is considered positive when he is 
north of the equator, and negative when south of it. 

Longitude is the distance east or west of any assumed meridian. The 
meridian passing through the Greenwich, England, observatory is the one 
commonly used. Longitude is measured 180° or twelve hours east and 
Avest of the assumed meridian. 

The jS'autical Almanac is a book, published by the government, which 
contains, among other data, the right ascensions and declinations of the 
sun, moon, planets, and certain fixed stars, the semi-diameters of the sun 
and moon, and the equation of time for Greenwich apparent and mean 
noon of each day in the year — all computed several years in advance. 

The folloAving is taken from the tables giving data in regard to the 
sun : 



JULY, 1896, AT GREENWICH APPARENT NOON. 



Day of the 
Week. 


o 5 

as 


Tlie Sun's 


Equation of 
Time to be 
added to Ap- 
])arent Time. 


S 5 

It 


Apparent 

Right 
Ascension. 


Differ- 
ence for 
One Hour 


Apparent 
Declination. 


Diflfer- 
ence for 
One Hour 


Semi- 
diameter. 


Wednesday, 
Thursday . 
Friday . . 

Saturday . 
SUNDAY. 
Monday . . 


1 

2 

3 

4 

5 
6 


h. m. s. 
6 43 49.95 
6 47 57.80 
6 52 5.38 

6 56 12.68 

7 19.65 
7 4 26.29 


s. 
10.332 
10.321 
10.310 

10.297 
10.284 
10.269 


o ' " 

N23 4 19.8 
22 59 48.2 
22 54 52.5 

22 49 32.7 
22 43 49.1 
22 37 41.7 


-10.81 
11.82 
12.82 

-13.82 
14.81 
15.80 


15 46.16 
15 46.15 
15 46.15 

15 46.15 
15 46.16 
15 46.17 


m. s. 
3 40.69 

3 51.95 

4 2.94 

4 13.64 
4 24.03 
4 34.09 


0.474 
0.463 
0.452 

0.439 
0.426 
0.412 



86 



ASTRONOMY AND NAVICxATION. 



The sextant shown in Fig. 35 is a hand instrument for measuring 
angles. Since it does not require a rigid support, it is particularly 
adapted to measuring angles from a ship at sea. 




The construction of the sextant depends upon the principle of optics, 
that if a ray of light be twice reflected from two plane mirrors, its angu- 
lar change in direction is equal to twice the angle of the mirrors. 

At / (Fig. 36) is the index glass which rotates with the arm IV. This 
arm carries at "F a vernier which moves along the graduated arc AB. At 
H is the horizon glass, one half of which is silvered and the other half clear. 




A ray of light coming from the direction ^S is reflected from I and H, and 
enters the eye at E. As the arm IV is moved along the arc, the image of S 



ASTRONOMY AND NAVIGATION. 87 

Wavels across the mirroi- at //. When tlie image is seen exactly in the 
line S'E, Av^hich is the path of a ray of light from ^S" through the clear 
part of the glass at H, the angle of the mirrors is one-half the angle SES'. 
The arc AB is so graduated that any angle, as SES', may be read directly 
by the vernier at V. 

To prove that SES' = twice the angle between the mirrors, draw DC 
perpendicular to the mirror at 7, and HC perpendicular to the mirror at 
H. Then DCH — angle between the mirrors. 

ZSES' =2x-2ir, 
ZDCH=x-y; 
.-. SES' ^2 DCH. 

If the two mirrors are not parallel when the vernier reads zero there 
is an index error. This error is a constant correction which must be 
made to all angles measured Avith the instrument. 

To determine the index error, measure the angular diameter of the 
sun by bringing the discs of the tAvo images in contact. 

Call the diameter of the sun d, the sextant reading r, and the index 
error e. 

Then d = r -{- e. 

Now move the vernier till the discs are again in contact, the image 
that was first above being now underneath. 

The zero of the vernier will now i robably be back of the zero of the 
circle. 

Call this reading — r'. 

Then — d = — r' + e. 

r' — r 
Hence e =^ 



In order to obtain the true altitude of a heavenly body, the observed 
altitude measured with the sextant must be corrected for dip, parallax, 
refraction, and, in the case of the sun or moon, for semi-diameter. 

Dip. 

Altitudes at sea are measured from the visible horizon, which is 
below the true horizon an amount depending upon the height of the 
observer above the surface of the Avater. 

Let OH (Fig. 37) be the true horizon, and OH the visible horizon. 
OA = a = the observer's height above the AA^ater. AC = BC = R -= mean 
radius of the earth =. 20,900,000 ft. approximately. HOH = ACB = D = 



38 



ASTRONOMY AND NAVIGATION. 



the dip of the horizon. Tan D = 



OB V2 Ba + a-' /2 a 



=v--+ 






IS 



BC R yi R R' R' 

very small and may be neglected. The angle D is also small, and may be 
taken as D tan 1'. Substituting, we have 

1 



D in minutes = 



tan 1' 3233 
= .94 Va, 



■y/a 




Pig. 37. 

or the dip in minutes is approximately the square root of the observer's 
height in feet above the water. 

The correction for dip must be subtracted from the observed altitude 
to give the true altitude. 

Parallax. 

The difference in direction of a heavenly body as seen by an observer 
on the earth's su.rface and as it would be seen from the earth's centre 
is called parallax. 

The magnitude of the parallax depends upon the altitude of the 
heavenly body and the ratio of the earth's radius to the distance of 
the heavenly body. The horizontal parallax is the parallax Avhen the 
heavenly body is in the horizon. It may be found as follows : 

B A 



Let P (Fig. 38) = horizontal parallax, AC = the earth's radius, BC 
the distance of the heavenly body from the earth's centre. 

AG 
BC 



SinP: 



ASTRONOMY AND NAVIGATION. 



39 



, The parallax of a heavenly body in any other position may be found 
as follows : Let ZAB (Fig. 39) be the observed zenith distance. ABC = p 
= parallax ; 




Fig. 39. 



then 



AC 
EG 



EC -.AC:: sin BAZ : sin ABC, 
sin BAZ X AC 



sin ABC = ' 



EC 



horizontal parallax. Substituting this, we have 



sinp = sin P sin BAZ. 

Both p and P are small ; therefore we may without excessive error assume 

p = Psm BAZ, 

or the parallax in any position is equal to the horizontal parallax into the 
sine of the observed zenith distance. 

The correction for parallax must be added to the observed altitude to 
give the true altitude. The parallax of the sun is small, never exceeding 
9", and in this work is neglected. 

Refraction. 

A ray of light coming obliquely through the atmosphere is bent out 
of a straight line so that the observer sees a heavenly body above its true 
position. 

Let SO (Fig. 40) be the direction of the ray before refraction, and OS" 
that of the refracted ray; then, from the laws of refraction, OS, OZ, OS" 

lie in the same plane, and is constant for the same media, 

^ ' sin Z'OS" 

whatever the angle SOZ. Call this ratio i; then sin (x + y)= i sin a;, 
sin X cos y + cos x sin y = i sin x. 



40 



ASTRONOMY AJfD NAVIGATION". 



The angle of refraction ij is small, and sin y = y approximately, and 
cos 2/ = 1 approximately. Substituting these values, 

sin X + y cos x = i sin x, y — (i — 1) tan x. 

Let Y equal the refraction when ZOS = 45*^ ; then taui^c — 1 and Y— i — 1. 
Substituting these values Ave have, 

y = Y tan x ; 

or the refraction for any zenith distance is equal to the product of refrac- 
tion at 45° into the tangent of the observed zenith distance. 



/s' 




z 

Fig. 40. 



AVith the barometer at 30 in. and the thermometer at 50° F., the 
refraction at 45° is 58.2". 

For all altitudes of 20° or over measured with a sextant or engineer's 
transit it is permissible to take the refraction correction in minutes equal 
to the natural tangent of the observed zenith distance. For altitudes less 
than 20° the correction obtained in this way will be too large. 

If the altitude of a heavenly body is measured with a sextant, the true 
altitude = observed altitude — dip + parallax — refraction. 

If the transit is used, there is no correction for dip. 

When the body observed is the sun or moon, the altitude of one limb 
is measured, and the angular semi-diameter must be added or subtracted 
according as the lower or the upper limb is observed. 



To find Latitude. 

First. By observation on a circumpolar star at upper or lower cul- 
mination. 

Let P (Fig. 41) represent the pole, S' a circumpolar star at upper 
and S a circumpolar star at lower culmination. 



ASTROXOMY AND XAVIGATIOX. 41 

, Let /;' = S'lr = the true altitude of ^', aud h the true altitude of S. 
Let d = Q'S' = the declination; then latitude = // +(90° — d) for lower 
culmination, and latitude = h' —(90° — d) for upper culmination. 

The latitude may also be found by observing, when possible, the alti- 
tude of a circumpolar star at both upper and lower culmination, and 
taking one-half the sum. 

Double latitude = h + (90°- d) + h'-~ (90°- d), or latitude = i (A + h'). 
Ill this case it is not necessary to know the declination. 




For the greater part of the year this observation cannot be taken, 
because one of the culminations occurs during daylight. 

Example. — The altitude of Polaris at upper culmination, as observed 
with an engineer's transit, was 45° 28'. Declination = 88° 45' 15"; what 
was the latitude ? 

45°28' = obs. alt. 
01 ' = ref r. cor. 



45° 27' = true altitude. 
88° 45' 15" 



134° 12' 15" 
90° 



44° 12' 15"N. = latitude. 

Second. By observing the meridian altitude of any star. 

In the northern hemisphere the meridian altitude of any star that cul- 
minates south of the zenith, minus its declination, is equal to the altitude 
of the equator. This is the complement of the latitude. If the star cul- 
minates north of the zenith, (180° — A) must be substituted for the altitude. 

Example. — The observed meridian altitude of the sun's lower limb, 
as measured from the visible horizon at sea, was 33° 22' 30"; sun's semi- 
diameter, 16'; height of eye above surface of water, 16 ft. ; sun's declina- 
tion south, 13° 03'. Required latitude. 



42 ASTRONOMY AXD NAVIGATION. 

33°22'30"obs. alt. 
1^30" ref r. cor. 
33° 21' 

04' cor. for dip. 
33° 17' 

16' sun's diam. 
33° 33' true altitude of sun's centre. 
— 13° 03' sun's declination. 
46° 36' = CO. latitude. 
43°24'N. = latitude. 

Third. By the altitude of the sun or a star in any position, the time 
being known. 

Z 




Fig. 42. 

This requires the solution of a spherical triangle. Let S (Fig. 42) 
represent the position of the sun or a star whose altitude has been 
measured. In the triangle ZSP, 

ZS = 90° - altitude, 
SP = 90° - declination, 
ZP = 90° - latitude, 

and the angle ZPS — 360° — time expressed in degrees. Therefore in the 
triangle ZSP we have two sides, and the angle opposite one of them given, 
to find the third side, which is the complement of the latitude required. 

Time. 

Apparent time is the time of the true sun. 

An apparent solar day is the interval of time between two successive 
transits of the sun over the same meridian. 

Because the motion of the earth about the sun is not uniform, and also 
becau.se it moves in the plane of the ecliptic instead of that of the equator, 
solar days are not of the same length, the greatest variation being about 
sixteen minutes. 



ASTRONOMY AND NAVIGATION. 43 

To obviate this cliificulty, an imaginary sun is supposed to revolve in 
the equator, about tlie earth, in the same time that the real sun appears 
to revolve in the ecliptic : this latter gives uniform motion. 

A mean solar day is the interval of time between two successive transits 
of this imaginary sun over the same meridian. The equation of time is the 
time that has to be added algebraically to apparent time to give mean time. 

The civil day begins at midnight, and is divided into two parts of 
twelve hours each. The astronomical day begins at noon of the civil day, 
and is divided into twenty-four hours. 

To change from civil time to astronomical time: If it is a.m. civil time, 
subtract one day and add twelve hours ; if it is p.m., omit the p.m. 

To change apparent time to mean solar time: Add algebraically the 
equation of time to apparent time, and the result is mean solar time. 

To determine local time by observing the altitude of the sun in the morn- 
ing or afternoon, the latitude being knoimi : In Pig. 42, let S represent 
the sun, the altitude of which has been measured, Z the zenith, and P the 
pole. In the triangle ZSP, SZ='dQ° - altitude, PS = 90°- declination, 
and PZ = 90° - latitude. 

Let h = altitude, I = latitude, d = declination, and z — zenith distance ; 

then sin i SPZ^J''"" ^ ^^ + (^ - ^)] '''' | [^ " (^ " ^R 
^ cos I COS d 

The angle SPZ is the hour angle if the altitude is measured in the 
afternoon. If the altitude is measured in the morning, the hour angle is 
360° minus the angle obtained by the solution of the triangle. The horn- 
angle changed to hours, minutes, and seconds is local apparent time, which 
may be changed to mean local time by adding the equation of time. 

Example. — Aug. 11, 1894, a.m. In latitude 42°30'Isr. the observed 
altitude of the sun's lower limb was 38° 19' ; height of eye, 25 ft. ; 
sun's declination, 15° 12' N. ; semi-diameter, 16'; equation of time, 



+ 5 "^■ 


01 '■. Required local time. 














42°30' = Z 






colog cos = 


0.1324 






15°12'-d 






COlog COS = 


0.0155 


38° 19' 




27° 18'= (I -d) 










5' 


dip. 


51° 32' = 2 










1' 


refr. cor. 


2)78° 50' 










38° 13' 




39° 25' = ^[z + (I - 


-d)l 




log sin = 


9.8027 


16' 


semi-diam. 


2)24° 14' 










38° 28' 


= h. 


12° 07' = ![.-(/ - 


-d)] 


log 


log sin = 
sin -1 SPZ = 

lspz= 


9.3220 
2)9.2726 

9.6363 
25° 38 '.9 



.SPZ =51° 17 '.8. 



44 ASTRONOMY AND NAVIGATION. 

360° - SPZ = 308° 42 '.2 = 20 h. 34 m. 49 s. 

_ 3 h. 3^ 111. ^g s. ^^ apparent time 
5 01 eq. of t. 

8 ^- 39 "'• 50 '• A.M. mean local time. 

Ans. 

Aug. 6, 1894, P.M. In latitude 42° 30' IST. the observed altitude of the 
sun's upper limb was 32° 55'; height of eye, 16 ft.; sun's declination, 
16°32']Sr. ; semi-diameter, 16'; equation of tin^e, +5 "'•39'-. Kequired 
mean local time. 4 ''• 08 '"• 27 ^•. Ans. 

To establish a Meridian Line by an Observation on the North Star 
(Polaris) at Elongation. 

Polaris appears to revolve about the pole in a sm :]1 circle in a little 
less than twenty-four hours. When at its extreme east or west point, it 
is said to be- at elongation. Fifteen or twenty minutes before the star 
gets to its elongation, set up the transit at one end of the line to be es- 
tablished and sight on the star. The star will move toward the east if 
the elongation is east, and toward the west if the elongation is west. 
When the star reaches its greatest elongation, it will move vertically along 
the cross-hair for some time, and then leave it in a direction opposite to 
its former motion. As long as the star moves toward the point of its 
elongation, the cross-hair mu.st be kept sighted on it, by turning the tan- 
gent screw. When the point of elongation is reached, the line defined by 
the transit must be established on the ground, by driving a stake or spike 
in this line at some point 100 to 200 ft. from the transit. 

To establish a meridian, lay off from this line the azimuth of Polaris, 
to the ricjlit if west elongation was observed, and to the left if east elon- 
gation was observed, the transit being set over the south end of the line. 

To find Azimuth. 

,1 sin polar distance 
sm azimuth = * ^ 

cos latitude 

It is necessary to luminate the cross-hairs when observing at night. 
This is done by means of a lantern, and a reflector made to fit over the 
object-glass end of the telescope. 

A temporary reflector may be made by fastening over the object-glass 
end of the telescope a piece of oiled paper, leaving a hole one-fourth to 
one-half inch in diameter over the centre of the object-glass. 

The times at which the elongations of Polaris occur are given in Bul- 
letin No. 14, published by the United States Coast and Geodetic Survey, 
at Washington. 



NAVIGATION. 



45 



NAVIGATION. 

There are two ways by which the navigator determines the position of 
his vessel at sea : by dead reckoning and by observation. 

In navigation by dead reckoning the courses and distances run are 
measured by tlie compass and log. 

In navigation by observation the position is determined by observing 
the altitude of the sun or some other heavenly body. 

The mariner's compass, shown in Fig. 43, consists of a circular card 
divided into thirty-two equal parts called points. Attached to the under 




side of the card, in the direction of its north and south line, is a mag- 
netic needle, supported at its centre on a pivot. Both swing in gimbals 
in a box, so that the card remains horizontal whatever the inclination of 
the ship. On the inside of the box are two points, o and h, the line con- 
necting which is parallel to the ship's keel ; so that if a is toward the 
bow, the point on the card opposite it will give the ship's course. 

Each of the thirty-two points are subdivided into quarter-points. 
Naming the points in order from the north throi;gh the east, south, and 
west, is called boxing the compass. 

The compass gives the magnetic course. In order to determine the true 
course, the magnetic course must be corrected for variation and deviation. 



46 



NAVIGATION. 



Variation of the mariner's compass is the angle that the needle makes 
with the true meridian. It is the same as declination of the surveyor's 
compass. 

Deviation is the change in the direction of the pointing of the needle 
caused by the iron and steel used in constructing the ship. Deviation 
changes with the direction in which the ship is headed. It is generally 
counteracted as much as possible by placing permanent magnets near the 
compass. 

Leeway is the angle which a vessel's course makes with her keel. It 
is caused by the wind sliding the vessel to leeward. 

Although leeway is not a compass error, the effect is the same as if it 
were, and the compass course must be corrected for it, to give the true 
course of the ship. 

When the variation is , , the true course is to the ■ i , of the corn- 
east' right 

pass course. The same rule applies in correcting for deviation. 

The correction for leeway on the , tack is the same as for 

■^ port east 



variation. 

A ship is on the , tack when the wind blows on the 

^ port 

of the ship. 



m1* ^'^^ 



Compass Course. 


Variatiou. 


Deviation. 


Leeway. 


True Course. 


W.S.W. 
N. by E. 

N.W.fN. 


ipt.W. 
Ipt.E. 
2^pts.W. 


ipt.W. 

iPt.w. 

ipt.E. 


1 pt. port. 
|- pt. starb. 
\ pt. starb. 


S.W.byW.iW. 
N. by E. 1- E. 
W.N.W. 



The common log consists of a reel, line, and chip. The chip (Fig. 44) is 
a flat piece of wood shaped like a sector of a circle, weighted with lead on its 
curved edge to make it stand upright in the water. A jerk on the line 




Fig. 44. 



will free the two lower cords at the toggle, causing the chip to lie fiat, so 
that it may be easily hauled aboard. With this log a sand-glass is used 
to measure the time. The line is graduated to the same part of a knot as 



NAVIGATION. 47 

the part of an hour measured by the sand-glass. This makes the knot 
47' 4" long when a twenty-eight-second glass is used. Vessels going at 
high speed use a fourteen-second glass. 

The patent log is an instrument fitted with curved blades, shaped 
somewhat like a ship's propeller, which cause it to revolve when drawn 
through the water. The number of revolutions are registered on a dial, 
which is graduated so as to record the revolutions per minute in knots per 
hour. 

The log gives the speed of the vessel in relation to the water. If 
there are currents, then the speed indicated by the log must be corrected ; 
i.e. if sailing against a current, deduct its speed from that given by the 
log ; if sailing with the current, add its speed. 

In a head wind the log is apt to overrate, and vice versa. 

A knot is equal to one minute (!') of latitude, or to one minute of 
longitude at the equator. Thus if a vessel sails due N. 60 knots from 
latitude 40° S., she will arrive in latitude 39° S. Since all meridians con- 
verge, a knot is equal to a minute of longitude only at the equator. 

The difference in longitude in minutes of angle may, however, be found 
when sailing due E. or W., by dividing the distance in knots by the cosine 
of the latitude. 



Fig. 45. 

Let P (Fig. 45) be the pole, EQ an arc of the equator, XFa parallel in 

latitude EX- then ZECX = ACXD = latitude = i. Cos CXD = ^; or 

DX ^ 

cos L = — —• R = radius. 
B 

Since similar arcs are to each other as their radii, 

EQ: XY:: R : R cos L. 

XY 

Hence EQ = — But EQ is the difference in longitude, and XY is 

cos L 

the distance sailed E. or W. 



48 



NAVIGATION. 



A ship sails 55 knots due E. from latitude 40, longitude 65 W. Find 
her longitude. 

55 



.766 



72'=1°12'. 65° -1° 12' = 63° 48' W. 



Ans. 



Middle Latitude Sailing. 

If a vessel sails on any course other than a meridian or parallel 
the new position may be found by computing the latitude and de- 
parture in the same way that they are computed in compass surveys. 
The latitude gives the distance sailed N. or S. of the starting-point, 
and the departure the distance E. or W. In like manner, if a vessel 
sails on several courses, the algebraic sum of all the latitudes gives 
the distance sailed IST. or S., and the algebraic sum of all the departures 
gives the distance sailed E. or W. If the distances are in knots, the 
algebraic sum of the latitudes is the difference of latitude in minutes of 
angle. To determine the difference in longitude, the algebraic sum of the 
departures must be divided by the cosine of the latitude ; but the course 
sailed Avas not on any one parallel of latitude. We may, however, deter- 
mine the difference in longitude by dividing the departure by the cosine of 
the mean of the latitudes of the two ends of the course ; or, in other 
words, by dividing the departure by the cosine of the middle latitude. This 
will give the difference in longitude somewhat less than the true difference. 
In low latitudes, where the difference in latitude of the two ends of the 
course is not great, the error is small. 

A vessel sails from lat. 40° 28' N., long. 74° 01' W., as follows : 

E. by N. . . . . 10 knots, 

E.S.E 20 " - 

S.E. by E 26 " 

S.S.W 15 " 

Require her lat. and long. 



Course. 


Angle. 


j Distance. 
Knots. 

1 


Latitude. 


Departure. 


N. 


s. 


E. W. 


E. by N. 


78° 45' 


1 10 


1.95 




9.81 




E.S.E. 


67° 30' 


1 20 




14.44 


21.62 





S.E. by E. 


56° 15' 


: 26 




7.65 


18.48 


S.S.W. 


22° 30' 


i 15 




13.86 


j 5.74 








1.95 


35.95 


49.91 


5.74 






Diff. in lat. 


in miu. = 


1.95 


5.74 




= 34.00 


44.17 





NAVIGATION. 49 

40° 28' - 34' = 39° 54' K = latitude. 

40° 11' ^ middle latitude, cos = .7640. 

— '- — = 58' = diff. in lonsr. in minutes. 
.7640 

74° 01' - 58' = 73° 03' W. = longitude. 

A chronometer is a well-made and nicely adjusted timepiece swung in 
gimbals and set to Greenwich time. 

The rate of a chronometer is the amount of time that it gains or loses 
in a day. 

The difference between the chronometer time and Greenwich time and 
the rate of the chronometer are determined when in port. Knowing these, 
the Greenwich time may be determined, when desired, during the voyage. 

On June 12 a chronometer compared with Greenwich time was 2 '"■ 14 '■ 
fast ; its rate was 1.1 -'• losing. What was the Greenwich time correspond- 
ing to 4 ^- 40 ""■ 17 ^- June 20 ? 

8 X 1.1^=9'-. 2™14^-9'=2"'-5% 
4h. 40 m. 17 s. _ 2 m. 5 s- ^ 4 h- 38 "^- 12 ^- Greenwich time. 

Since position determined by dead reckoning is always liable to error, 
it is customary to determine the position of a ship by observation, daily if 
possible. 

In determining the position of a ship by observation, the latitude is 
found by measuring the meridian altitude of the sun, as explained in the 
chapter on astronomy. 

The longitude is determined by measuring the altitude of the sun 
when it bears nearly E. or W., and computing local time. The ship's 
chronometer gives Greenwich time. The difference between these is the 
difference in longitude. The longitude is W. if the Greenwich time is 
later than the local time, and E. if it is earlier. 

The change in latitude during the interval of time between the obser- 
vation for latitude and that for time is found by dead reckoning. It is 
necessary to compute this in order to get the latitude to be used in solving 
for time. 

On Eeb. 28, 1896, at 21''-40'"- approximate Greenwich time, the 
observed meridian altitude of the sun's lower limb was 33° 36' 30", the 
sun bearing south. Height of eye above the sea 20 ft. ; sun's declination 
7° 41' 30" S. ; semi-diameter 16'. Required the latitude. 48° 32' X. Ans. 

On May 20, 1896, at 22 1^- 20"" approximate Greenwich time, the 
observed meridian altitude of the sun's lower limb was 68° 12' 30", the 



50 NAVIGATION. 

sun bearing south. Height of eye 25 ft.; sun's declination 20°20']Sf. ; 
semi-cliameter 16'. Required the latitude. 41°57'N. Ans. 

On Sept. 29, 1896, at 0'' 20"'- approximate Greenwich time, the 
observed meridian altitude of the sun's upper limb was 63° 25', the sun 
bearing north. Height of eye 20 ft.; san's declination 2°43'S. ; semi- 
diameter 16'. Required the latitude. 29°39'S. A^is. 

On March 11, 1896, a.m., in latitude 47° 28' N., the observed altitude 
of the sun's lower limb was 26° 25'. The corresponding Greenwich time 
was 23'^- 31™- 14 \ Height of eye 25 ft.; sun's declination 3° 24' S. ; 
semi-diameter 16'; equation of time + 9"'- 59'-. Required the longitude. 

2 ''■ 14 "'• 30 ^- W. A71S. 

On Sept. 6, 1896, in latitude 36° 18' N., the observed altitude of the 
sun's lower limb was 33° 14'. The corresponding Greenwich time was 
lj\,. 20"'- 41 ^ Height of eye 20 ft.; sun's declination 6°16'N.; semi- 
diameter 16'; equation of time — 1 "'• 51 '■. Required the longitude. 

10 ^- 08 '"• 08 ^' E. Ans. 



STADIA TABLES.i 



M. 


0° 


10 


2° 


3° 




hor. dist. 


ditf. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. ■ 


diff. elev. 


0' 


100.00 


0.00 


99.97 


1.74 


99.88 


3.49 


99.73 


5.23 


2 


100.00 


0.06 


99.97 


1.80 


99.87 


3.55 


99.72 


5.28 


4 


100.00 


0.12 


99.97 


1.86 


99.87 


3.60 


99.71 


5.34 


6 


100.00 


0.17 


99.96 


1.92 


99.87 


3.66 


99.71 


5.40 


8 


100.00 


0.23 


99.96 


1.98 


99.86 


3.72 


99.70 


5.46 


10 


100.00 


0.29 


99.96 


2.04 


99.86 


3.78 


99.69 


5.52 


12 


100.00 


0.35 


99.96 


2.09 


99.85 


3.84 


99.69 


5.57 


14 


100.00 


0.41 


99.95 


2.15 


99.85 


3.90 


99.68 


5.63 


16 


100.00 


.0.47 


99.95 


2.21 


99.84 


3.95 


99.68 


5.69 


18 


100.00 


0.52 


99.95 


2.27 


99.84 


4.01 


99.67 


5.75 


20 


100.00 


0.58 


99.95 


2.33 


99.83 


4.07 


99.66 


5.80 


22 


100.00 


0.64 


99.94 


2.38 


99.83 


4.13 


99.66 


5.86 


24 


100.00 


0.70 


99.94 


2.44 


99.82 


4.18 


99.65 


5.92 


26 


99.99 


0.76 


99.94 


2.50 


99.82 


4.24 


99.64 


5.98 


28 


99.99 


0.81 


99.93 


2.56 


99.81 


4.30 


99.63 


6.04 


30 


99.99 


0.87 


99.93 


2.62 


99.81 


4.36 


99.63 


6.09 


32 


99.99 


0.93 


99.93 


2.67 


99.80 


4.42 


99.62 


6.15 


34 


99.99 


0.99 


99.93 


2.73 


99.80 


4.48 


99.62 


6.21 


36 


99.99 


1.05 


99.92 


2.79 


99.79 


4.53 


99.61 


6.27 


38 


99.99 


1.11 


99.92 


2.85 


99.79 


4.59 


99.60 


6.33 


40 


99.99 


1.16 


99.92 


2.91 


99.78 


4.65 


99.59 


6.38 


42 


99.99 


1.22 


99.91 


2.97 


99.78 


4.71 


99.59 


6.44 


44 


99.98 


1.28 


99.91 


3.02 


99.77 


4.76 


99.58 


6.50- 


46 


99.98 


1.34 


99.90 


3.08 


99.77 


4.82 


99.57 


6.56 


48 


99.98 


1.40 


99.90 


3.14 


99.76 


4.88 


99.56 


6.61 


50 


99.98 


1.45 


99.90 


3.20 


99.76 


4.94 


99.56 


6.67 


52 


99.98 


1.51 


99.89 


3.26 


99.75 


4.99 


99.55 


6.73 


54 


99.98 


1.57 


99.89 


3.31 


99.74 


5.05 


99.54 


6.78 


56 


99.97 


1.63 


99.89 


3.37 


99.74 


5.11 


99.53 


6.84 


58 


99.97 


1.69 


99.88 


3.43 


99.73 


5.17 


99.52 


6.90 


60 
c = 0.75 
c = 1.00 
c = 1.25 


99.97 


1.74 


99.88 


3.49 


99.73 


5.23 


99.51 


6.96 


0.75 


0.01 


0.75 


0.02 


0.75 


0.03 


0.75 


0.05 


1.00 


0.01 


1.00 


0.03 


1.00 


0.04 


1.00 


0.06 


1.25 


0.02 


1.25 


0.03 


1.25 


0.05 


1.25 


0.08 



1 These tables were computed by Mr. Arthur AVinslow of the State Geological 
Survey of Pennsylvania. 

51 



52 



STADIA TABLES. 



M, ^ 


40 








6° 


7 







hor. dist. 


diif. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diiT. elev. 


0' 


99.51 


6.96 


99.24 


8.68 


98.91 


10.40 


98.51 


12.10 


2 


99.51 


7.02 


99.23 


8.74 


98.90 


10.45 


98.50 


12.15 


4 


99.50 


7.07 


99.22 


8.80 


98.88 


10.51 


98.48 


12.21 


6 


99.49 


7.13 


99.21 


8.85 


98.87 


10.57 


98.47 


12.26 


8 


99.48 


7.19 


99.20 


8.91 


98.86 


10.62 


98.46 


12.32 


10 


99.47 


7.25 


99.19 


8.97 


98.85 


10.68 


98.44 


12.38 


12 


99.46 


7.30 


99.18 


9.03 


98.83 


10.74 


98.43 


12.43 


14 


99.46 


7.36 


99.17 


9.08 


98.82 


10.79 


98,41 


12.49 


IG 


99.45 


7.42 


99.16 


9.14 


98.81 


10.85 


98.40 


12.55 


18 


99.44 


7.48 


99.15 


9.20 


98.80 


10.91 


98.39 


12.60 


20 


99.43 


7.53 


99.14 


9.25 


98.78 


10.96 


98.37 


12.66 


22 


99.42 


7.59 


99.13 


9.31 


98.77 


11.02 


98.36 


12.72 


24 


99.41 


7.65 


99.11 


9.37 


98.76 


11.08 


98.34 


12.77 


26 


99.40 


7.71 


99.10 


9.43 


98.74 


11.13 


98.33 


12.83 


28 


99.39 


7.76 


99.09 


9.48 


98.73 


11.19 


98.31 


12.88 


30 


99.38 


7.82 


99.08 


9.54 


98.72 


11.25 


98.29 


12.94 


32 


99.38 


7.88 


99.07 


9.60 


98.71 


11.30 


98.28 


13.00 


34 


99.37 


7.94 


99.06 


9.65 


98.69 


11.36 


98.27 


13.05 


36 


99.36 


7.99 


99.05 


9.71 


98.68 


11.42 


98.25 


13.11 


38 


99.35 


8.05 


99.04 


9.77 


98.67 


11.47 


98.24 


13.17 


40 


99.34 


8.11 


99.03 


9.83 


98.65 


11.53 


98.22 


13.22 


42 


99.33 


8.17 


99.01 


9.88 


98.64 


11.59 


98.20 


13.28 


44 


99.32 


8.22 


99.00 


9.94 


98.63 


11.64 


98.19 


13.33 


46 


99.31 


8.28 


98.99 


10.00 


98.61 


11.70 


98.17 


13.39 


48 


99.30 


8.34 


98.98 


10.05 


98.60 


11.76 


98.16 


13.45 


50 


99.29 


8.40 


98.97 


10.11 


^- 98.58 


11.81 


98.14 


13.50 


52 


99.28 


8.45 


98.96 


10.17 


98.57 


11.87 


98.13 


13.56 


54 


99.27 


8.51 


98.94 


10.22 


98.56 


11.93 


98.11 


13.61 


56 


99.26 


8.57 


98.93 


10.28 


98.54 


11.98 


98.10 


13.67 


58 


99.25 


8.63 


98.92 


10.34 


98.53 


12.04 


98.08 


13.73 


60 
c = 0.75 
c = 1.00 
c = 1.25 


99.24 


8.68 


98.91 


10.40 


98.51 


12.10 


98.06 


13.78 


0.75 


0.06 


0.75 


0.07' 


0.75 


0.08 


0.74 


0.10 


1.00 


0.08 


0.99 


0.09 


0.99 


0.11 


0.99 


0.13 


1.25 


0.10 


1.24 


0.11 


1.24 


0.14 


1.24 


0.16 



STADIA TABLES. 



53 



M. 


8° 


9° 


10° 


11° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


0' 


98.06 


13.78 


97.55 


15.45 


96.98 


17.10 


96.36 


18.73 


2 


98.05 


13.84 


97.53 


15.51 


96.96 


17.16 


96.34 


18.78 


4 


98.03 


13.89 


97.52 


15.56 


96.94 


17.21 


96.32 


18.84 


6 


98.01 


13.95 


97.50 


15.62 


96.92 


17.26 


96.29 


18.89 


8 


98.00 


14.01 


97.48 


15.67 


96.90 


17.32 


96.27 


18.95 


10 


97.98 


14.06 


97.46 


15.73 


96.88 


17.37 


96.25 


19.00 


12 


97.97 


14.12 


97.44 


15.78 


96.86 


17.43 


96.23 


19.05 


14 


97.95 


14.17 


97.43 


15.84 


96.84 


17.48 


96.21 


19.11 


16 


97.93 


14.23 


97.41 


15.89 


96.82 


17.54 


96.18 


19.16 


18 


97.92 


14.28 


97.39 


15.95 


96.80 


17.59 


96.16 


19.21 


20 


97.90 


14.34 


97.37 


16.00 


96.78 


17.65 


96.14 


19.27 


22 


97.88 


14.40 


97.35 


16.06 


96.76 


17.70 


96.12 


19.32 


24 


97.87 


14.45 


97.33 


16.11 


96.74 


17.76 


96.09 


19.38 


26 


97.85 


14.51 


97.31 


16.17 


96.72 


17.81 


96.07 


19.43 


28 


97.83 


14.56 


97.29 


16.22 


96.70 


17.86 


96.05 


19.48 


30 


97.82 


14.62 


97.28 


16.28 


96.68 


17.92 


96.03 


19.54 


32 


97.80 


14.67 


97.26 


16.33 


96.66 


17.97 


96.00 


19.59 


34 


97.78 


14.73 


97.24 


16.39 


96.64 


18.03 


95.98 


19.64 


36 


97.76 


14.79 


97.22 


16.44 


96.62 


18.08 


95.96 


19.70 


38 


97.75 


14.84 


97.20 


16.50 


96.60 


18.14 


95.93 


19.75 


40 


97.73 


14.90 


97.18 


16.55 


96.57 


18.19 


95.91 


19.80 


42 


97.71 


14.95 


97.16 


16.61 


96.55 


18.24 


95.89 


19.86 


44 


97.69 


15.01 


97.14 


16.66 


96.53 


18.30 


95.86 


19.91 


46 


97.68 


15.06 


97.12 


16.72 


96.51 


18.35 


95.84 


19.96 


48 


97.66 


15.12 


97.10 


16.77 


96.49 


18.41 


95.82 


20.02 


50 


97.64 


15.17 


97.08 


16.83 


96.47 


18.46 


95.79 


20.07 


52 


97.62 


15.23 


97.06 


16.88 


96.45 


18.51 


95.77 


20.12 


54 


97.61 


15.28 


97.04 


16.94 


96.42 


18.57 


95.75 


20.18 


56 


97.59 


15.34 


97.02 


16.99 


96.40 


18.62 


95.72 


20.23 


58 


97.57 


15.40 


97.00 


17.05 


96.-38 


18.68 


95.70 


20.28 


60 
c = 0.75 
c = 1.00 
c = 1.25 


97.55 


15.45 


96.98 


17.10 


96.-36 


18.73 


95.68 


20.34 


0.74 


0.11 


0.74 


0.12 


0.74 


0.14 


0.73 


0.15 


0.99 


0.15 


0.99 


0.16 


0.98 


0.18 


0.98 


0.20 


1.23 


0.18 


1.23 


0.21 


1.23 


0.23 


1.22 


0.25 



54 



STADIA TABLES. 



M. 


1 


2° 


130 


140 


i; 


o 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


0' 


95.68 


20.34 


94.94 


21.92 


94.15 


23.47 


93.30 


25.00 


2 


95.65 


20.39 


94.91 


21.97 


94.12 


23.52 


93.27 


25.05 


4 


95.63 


20.44 


94.89 


22.02 


94.09 


23.58 


93.24 


25.10 


6 


95.61 


20.50 


94.86 


22.08 


94.07 


23.63 


93.21 


25.15 


8 


95.58 


20.55 


94.84 


22.13 


94.04 


23.68 


93.18 


25.20 


10 


95.56 


20.60 


94.81 


22.18 


94.01 


23.73 


93.16 


25.25 


12 


95.53 


20.66 


94.79 


22.23 


93.98 


23.78 


93.13 


25.30 


U 


95.51 


20.71 


94.76 


22.28 


93.95 


23.83 


93.10 


25.35 


16 


95.49 


20.76 


94.73 


22.34 


93.93 


23.88 


93.07 


25.40 


18 


95.46 


20.81 


94.71 


22.39 


93.90 


23.93 


93.04 


25.45 


20 


95.44 


20.87 


94.68 


22.44 


93.87 


23.99 


93.01 


25.50 


22 


95.41 


20.92 


94.66 


22.49 


93.84 


24.04 


92.98 


25.55 


21 


95.39 


20.97 


94.63 


22.51 


93.81 


24.09 


92.95 


25.60 


26 


95.36 


21.03 


94.60 


22.60 


93.79 


24.14 


92.92 


25.65 


28 


95.34 


21.08 


94.58 


22.65 


93.76 


24.19 


92.89 


25.70 


30 


95.32 


21.13 


94.55 


22.70 


93.73 


24.24 


92.86 


25.75 


32 


95.29 


21.18 


94.52 


22.75 


93.70 


24.29 


92.83 


25.80 


34 


95.27 


21.24 


94.50 


22.80 


93.67 


24.34 


92.80 


25.85 


36 


95.24 


21.29 


94.47 


22.85 


93.65 


24.39 


92.77 


25.90 


38 


95.22 


21.34 


94.44 


22.91 


93.62 


24.44 


92.74 


25.95 


40 


95.19 


21.39 


94.42 


22.96 


93.59 


24.49 


92.71 


26.00 


42 


95.17 


21.45 


94.39 


23.01 


93.56 


24.55 


92.68 


26.05 


44 


95.14 


21.50 


94.36 


23.06 


93.53 


24.60 


92.65 


26.10 


46 


95.12 


21.55 


94.34 


23.11 


93.50 


.25.65 


92.02 


26.15 


48 


95.09 


21.60 


94.31 


23.16 


93.47 


24.70 


92.59 


26.20 


50 


95.07 


21.66 


94.28 


23.22 


93.45 


24.75 


92.56 


26.25 


52 


95.04 


21.71 


94.26 


23.27 


93.42 


24.80 


92.53 


26.30 


54 


95.02 


21.76 


94.23 


23.32 


93.39 


24.85 


92.49 


26.35 


56 


94.99 


21.81 


94.20 


23.37 


93.36 


24.90 


92.46 


26.40 


58 


94.97 


21.87 


94.17 


23.42 


93.33 


24.95 


92.43 


26.45 


60 
c ^ 0.75 
e = 1.00 
c = 1.25 


94.94 


21.92 


94.15 


23.47 


93.80 


25.00 


92.40 


26.50 


0.73 


0.16 


0.73 


0.17 


0.73 


0.19 


0.72 


0.20 


0.98 


0.22 


0.97 


0.23 


0.97 


0.25 


0.90 


0.27 


1.22 


0.27 


1.21 


0.29 


1.21 


0.31 


1.20 


0.34 



STADIA TABLES. 



55 



M. 


160 


17 





IS 





liio 




hor. dist. 


diff. elev. 


hor. dist. 


ditt-. t'lev. 


hor. dist. 


diff. elev. } 


hor. dist. 


diff. elev. 


0' 


92.40 


26.50 


91.45 j 


27.96 


90.45 


29.39 


89.40 


30.78 


2 


92.37 


20.55 


91.42 


28.01 


90.42 


29.44 


89.36 


30.83 


4 


92.34 


26.59 


91.39 


28.06 


90.38 


29.48 


89.32 


30.87 


6 


92.31 


26.64 


91.35 


28.10 


90.35 


29.53 


89.29 


30.92 


8 


92-28 


26.69 


91.32 


28.15 


90.31 


29.-58 


89.26 


30.97 


10 


92.25 


26.74 


91.29 


28.20 


90.28 


29.62 


89.22 


31.01 


12 


92.22 


26.79 


91.26 


28.25 


90.24 


29.67 


89.18 


31.06 


14 


92.19 


26.84 


91.22 


28.. 30 


90.21 


29.72 


89.15 


31.10 


16 


92.15 


26.89 


91.19 


28.34 


90.18 


29.76 


89.11 


31.15 


18 


92.12 


26.94 


91.16 


28.39 


90.14 


29.81 


89.08 


31.19 


20 


92.09 


26.99 


91.12 


28.44 


90.11 


29.86 


89.04 


31.24 


22 


92.06 


27.04 


91.09 


28.49 


90.07 


29.90 


89.00 


31.28 


24 


92.03 


27.09 


91.06 


28.54 


90.04 


29.95 


88.96 


31.33 


26 


92.00 


27.13 


91.02 


28.58 


90.00 


30.00 


88.93 


31.38 


28 


91.97 


27.18 


90.99 


28.63 


89.97 


30.04 


88.89 


31.42 


30 


91.93 


27.23 


90.96 


28.68 


89.93 


30.09 


88.86 


31.47 


32 


91.90 


27.28 


90.92 


28.73 


89.90 


30.14 


88.82 


31.51 


34 


91.87 


27-33 


90.89 


28.77 


89.86 


30.19 


88.78 


31.56 


36 


91.84 


27.38 


90.86 


28.82 


89.83 


30.23 


88.75 


31.60 


38 


91.81 


27.43 


90.82 


28.87 


89.79 


.30.28 


88.71 


31.65 


40 


91.77 


27.48 


90.79 


28.92 


89.76 


30.32 


88.67 


31.69 


42 


91.74 


27.52 


90.76 


28.96 


89.72 


30.37 


88.64 


31.74 


44 


91.71 


27.-57 


90.72 


29.01 


89.69 


30.41 


88.60 


31.78 


46 


91.68 


27.62 


90.69 


29.06 


89.65 


30.46 


88.56 


31.83 


48 


91.65 


27.67 


90.66 


29.11 


89.61 


30.51 


88.53 


31.87 


50 


91.61 


27.72 


90.62 


29.15 


89.58 


-30.55 


88.49 


31.92 


52 


91.58 


27.77 


90.59 


29.20 


89.54 


30.60 


88.45 


31.96 


54 


91.55 


27.81 


90.55 


29.25 


89.51 


30.65 


88.41 


32.01 


56 


91.52 


27.86 


90.52 


29.30 


89.47 


30.69 


88.-38 


32.05 


58 


91.48 


27.91 


90.48 


29.34 


89.44 


.30.74 


88..34 


32.09 


60 
c = 0.75 
c = 1.00 
c = 1.25 


91.45 


27.96 


90.45 


29.39 


89.40 


30.78 


88.30 


32.14 


0.72 


0.21 


0.72 


0.23 


0.71 


0.24 


0.71 


0.25 


1 0.96 


0.28 


0.95 


0.30 


0.95 


0.32 


0.94 


0.33 


1.20 


0.36 


1.19 


0.38 


1.19 


0.40 


1.18 


0.42 



56 



STADIA TABLES. 



M. 


20° 


21 


[O 


22° 


21 


,0 




hor. dist. 


diflF. elev. 


hor. dist. 


diflF. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


0' 


88.30 


32.14 


87.16 


33.46 


85.97 


34.73 


84.73 


35.97 


2 


88.26 


32.18 


87.12 


33.50 


85.93 


34.77 


84.69 


36.01 


4 


88.23 


32.23 


87.08 


33.54 


85.89 


34.82 


84.65 


3().05 


6 


88.19 


32.27 


87.04 


33.59 


85.85 


34.86 


84.61 


36.09 


8 


88.15 


32.-32 


87.00 


33.63 


85.80 


34.90 


81.57 


36.13 


10 


88.11 


32.36 


86.96 


33.67 


85.76 


34.94 


84.52 


36.17 


12 


88.08 


32.41 


86.92 


.33.72 


85.72 


34.98 


84.48 


36.21 


14 


88.04 


32.45 


86.88 


33.76 


85.68 


35.02 


84.44 


36.25 


16 


88.00 


32.49 


86.84 


33.80 


85.64 


35.07 


84.40 


36.29 


18 


87.96 


32.54 


86.80 


33.84 


85.60 


35.11 


84.35 


36.33 


20 


87.93 


32.58 


86.77 


33.89 


85.56 


35.15 


84.31 


36.37 


22 


87.89 


32.63 


86.73 


33.93 


85.52 


35.19 


84.27 


36.41 


24 


87.85 


32.67 


86.69 


33.97 


85.48 


35.23 


84.23 


36.45 


26 


87.81 


.32.72 


86.65 


34.01 


85.44 


35.27 


84.18 


36.49 


28 


87.77 


32.76 


86.61 


34.06 


85.45 


35.31 


84.14 


36.53 


30 


87.74 


32.80 


86.57 


34.10 


85.36 


35.36 


84.10 


36.57 


32 


87.70 


32.85 


86.53 


34.14 


85.31 


35.40 


84.06 


36.61 


34 


87 66 


32.89 


86.49 


34.18 


85.27 


35.44 


84.01 


36.65 


36 


87.62 


32.93 


86.45 


34.23 


85.23 


35.48 


83.97 


36.69 


38 


87.58 


32.98 


86.41 


34.27 


85 19 


35.52 


83.93 


36.73 


40 


87.54 


33.02 


86.37 


34.31 


85.15 


35.56 


83.89 


36.77 


42 


87.51 


33.07 


86.33 


34.35 


85.11 


35.60 


83.84 


36.80 


44 


87.47 


33.11 


86.29 


34.40 


85.07 


35.64 


83.80 


36.84 


46 


87.43 


33.15 


86.25 


34.44 


85.02 


3.3.68 


83.76 


3(;.88 


48 


87.39 


33.20 


86.21 


34.48 


81.98 


35.72 


83.72 


36.92 


50 


87.35 


33.24 


86.17 


34.52 


84.94 


35.76 


83.67 


36.96 


52 


87.31 


33.28 


86.13 


34.57 


84.90 


35.80 


83.63 


37.00 


54 


87.27 


33.33 


86.09 


34.61 


84.86 


35.85 


83.59 


37.04 


56 


87.24 


33.37 


86.05 


34.65 


84.82 


35.89 


83.54 


37.08 


58 


87.20 


.33.41 


86.01 


34.69 


84.77 


35.93 


83.50 


37.12 


60 
c = 0.75 
c = 1.00 
c = 1.25 


87.16 


33.46 


85.97 


34.73 


84.73 


35.97 


83.46 


37.16 


0.70 


0.26 


0.70 


0.27 


0.69 


0.29 


0.69 


0..30 


0.94 


0.35 


0.93 


0.37 


0.92 


0.38 


0.92 


0.40 


1.17 


0.44 


1.16 


0.46 


1.15 


0.48 


1.15 


0.50 



STADIA TABLES. 



57 



M. 


■2i 


to 


•2. 


>o 


•_> 


)0 


., 


-0 




hor. dist. 


diff. elev. 


hor. dist. 


ditf. elev. 


hor. dist. 


diflf. elev. 


hor. dist. 


diff. elev. 


0' 


83.46 


37.16 


82.14 


38.30 


80.78 


39.40 


79.39 


40.45 


2 


83.41 


37.20 


82.09 


38.34 


80.74 


39.44 


79.34 


40.49 


4 


83.37 


37.23 


82.05 


38.38 


80.69 


39.47 


79.30 


40.52 


6 


83.33 


37.27 


82.01 


38.41 


80.65 


39.51 


79.25 


40.55 


8 


83.28 


37.31 


81.96 


38.45 


80.60 


39.54 


79.20 


40.. 59 


10 


83.24 


37.35 


81.92 


38.49 


80.55 


39.58 


79.15 


40.62 


12 


83.20 


37.39 


81.87 


38.53 


80.51 


39.61 


79.11 


40.66 


14 


83.15 


37.43 


81.83 


38.56 


80.46 


39.65 


79.06 


40.69 


16 


83.11 


37.47 


81.78 


38.60 


80.41 


39.69 


79.01 


40.72 


18 


83.07 


37.51 


81.74 


38.64 


80.37 


39.72 


78.96 


40.76 


20 


83.02 


37.54 


81.69 


38.67 


80.32 


39.76 


78.92 


40.79 


22 


82.98 


37.58 


81.65 


38.71 


80.28 


39.79 


78.87 


40.82 


24 


82.93 


37.62 


81.60 


38.75 


80.23 


39.83 


78.82 


40.86 


26 


82.89 


37.66 


81.56 


38.78 


80.18 


39.86 


78.77 


40.89 


28 


82.85 


37.70 


81.51 


38.82 


80.14 


39.90 


78.73 


40.92 


30 


82.80 


37.74 


81.47 


38.86 


80.09 


39.93 


78.68 


40.96 


32 


82.76 


37.77 


81.42 


38.89 


80.04 


39.97 


78.63 


40.99 


34 


82.72 


37.81 


81.38 


38.93 


80.00 


40.00 


78.58 


41.02 


36 


82.67 


37.85 


81.33 


38.97 


79.95 


40.04 


78.54 


41.06 


38 


82.63 


37.89 


81.28 


39.00 


79.90 


40.07 


78.49 


41.09 


40 


82.58 


37.93 


81.24 


39.04 


79.86 


40.11 


78.44 


41.12 


42 


82.54 


37.96 


81.19 


39.08 


79.81 


40.14 


78.39 


41.16 


44 


82.49 


38.00 


81.15 


39.11 


79.76 


40.18 


78.34 


41.19 


46 


82.45 


38.04 


81.10 


39.15 


79.72 


40.21 


78.30 


41.22^ 


48 


82.41 


38.08 


81.06 


39.18 


79.67 


40.24 


78.25 


41.26 


50 


82.36 


38.11 


81.01 


39.22 


79.62 


40.28 


78.20 


41.29 


52 


82.32 


38.15 


80.97 


39.26 


79.58 


40.31 


78.15 


41. .32 


54 


82.27 


38.19 


80.92 


39.29 


79.53 


40.35 


78.10 


41.35 


56 


82.23 


38.23 


80.87 


39..S3 


79.48 


40.38 


78.06 


41.39 


58 


82.18 


38.26 


80.83 


39.36 


79.44 


40.42 


78.01 


41.42 


60 
c = 0.75 
c = 1.00 
c = 1.25 


82.14 


38.30 


80.78 


39.40 


79.39 


40.45 


77.96 


41.45 


0.G8 


0.31 


0.68 


0.32 


0.67 


0.33 


0.66 


0.35 


0.9. 


0.41 


0.90 


0.43 


0.89 


0.45 


0.89 


0.46 


1.14 


0.52 


1.13 


0.54 


1.12 


0.56 


1.11 


0.58 



58 



STADIA TABLES. 



.M. 


28° 


29° 


30O 




hor. dist. 


dift". elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


0' 


77.96 


41.45 


76.50 


42.40 


75.00 


43.30 


2 


77.91 


41.48 


76.45 


42.43 


74.95 


43.33- 


4 


77.86 


41.52 


76.40 


42.46 


74.90 


43.36 


6 


77.81 


41.55 


76.35 


42.49 


74.85 


43.39 


8 


77.77 


41.58 


76.30 


42.53 


74.80 


43.42 


10 


77.72 


41.61 


76.25 


42.56 


74.75 


43.45 


12 


77.67 


41.65 


76.20 


42.59 


74.70 


43.47 


14 


77.62 


41.68 


76.15 


42.62 


74.65 


43.50 


16 


■77.57 


41.71 


76.10 


42.65 


74.60 


43.53 


18 


77.52 


41.74 


76.05 


42.68 


74.55 


43.56 


20 


77.48 


41.77 


76.00 


42.71 


74.49 


43.59 


22 


■ 77.42 


41.81 


75.95 


42.74 


74.44 


43.62 


24 


77.38 


41.84 


75.90 


42.77 


74.39 


43.65 


26 


77.33 


41.87 


75.85 


42.80 


74.34 


43.67 


28 


77.28 


41.90 


75.80 


42.83 


74.29 


43.70 


30 


77.23 


41.93 


75.75 


42.86 


74.24 


43.73 


32 


77.18 


41.97 


75.70 


42.89 


74.19 


43.76 


34 


77.13 


42.00 


75.65 


42.92 


74.14 


43.79 


36 


77.09 


42.03 


75.60 


42.95 


74.09 


43.82 


38 


77.04 


42.06 


75.55 


42.98 


74.04 


43.84 


40 


76.99 


42.09 


75-50 


43.01 


73.99 


43.87 


42 


76.94 


42.12 


75.45 


43.04 


73.93 


43.90 


44 


76.89 


42.15 


75.40 


43.07 


73.88 


43.93 


46 


76.84 


42.19 


75.35 


43.10 


73.83 


43.95 


48 


76.79 


42.22 


75.30 


43.13 


73.78 


43.98 


50 


76.74 


42.25 


75.25 


43.16 


73.73 


44.01 


52 


76.69 


42.28 


75.20 


43.18 


73.68 


44.04 


54 


76.64 


42.31 


75.15 


43.21 


73.63 


41.07 


56 


76.59 


42.34 


75.10 


43.24 


73.58 


44.09 


58 


76.55 


42.37 


75.05 


43.27 


73.52 


44.12 


GO 
c = 0.75 
c= 1.00 
c = 1.25 


76.50 


42.40 


75.00 


43.30 


73.47 


44.15 


0-66 


0.36 


0.65 


0.37 


0.65 


0.33 


0.88 


0.48 


0.87 


0.49 


0.83 


0.51 


1.10 


O.CO 


1.09 


0.62 


1.08 


0.64 



A TABLE OP MEAN REFKACTIONS IN DECLINATION. 

To apply on the declination arc of Solar Attachment of 
either Compasses or Transits. 

Computed by Edwabd W. Akms, C. E., for W. & L. E. Guklet. Troy, N. Y. 



O 


DECLmATIONS. 














< 






Foe Latitude 15°. 








o 

w 














+ 20° 


+ 15° 


+ 10° 


+ 5° 


0° 


-5° 


-10° 


-15^ 


-20" 


Oh. 


-05" 


0" 


+ 05" 


10" 


15" 


21" 


27" 


33" 


40" 


2 


-03 


+ 02 


07 


12 


18 


23 


29 


36 


43 


3 


+ 01 


05 


11 


16 


22 


28 


34 


41 


49 


4 


08 


12 


19 


24 


30 


37 


44 


53 


1'04 


5 


29 


34 


41 


49 


59 


I'lO 


1'24 


1'43 


2 08 


Fob Latitude 17° 30'. 


Oh. 


-02" 


+ 02" 


08" 


13" 


18" 


24" 


30" 


36" 


44" 


2 





05 


10] 


15 


21 


27 


33 


40 


48 


H 


+ 03 


10 


15 


21 


27 


33 


40 


48 


57 


4 


13 


18 


23 


29 


35 


43 


51 


I'Ol 


1'13 


5 


34 


41 


49 


58 


I'lO 


1'23 


1'41 


206 


242 


Fob Latitude 20". ' 


Oh. 


0" 


05" 


10" 


15" 


21" 


27" 


33" 


40" 


48" 


2 


03 


or 


13 


18 


24 


30 


36 


44 


52 


3 


06 


13 


18 


24 


30 


36 


44 


52 


1'02 


4 


17 


22 


28 


35 


42 


50 


I'CO 


I'll 


126 


5 


39 


47 


57 


1'07 


1'20 


1'37 


2 00 


2 32 


3 25 


Fob Latitude 22° 30'. 


Oh. 


02" 


08" 


13" 


18" 


24" 


36" 


36" 


44" 


52" 


2 


06 


11 


15 


21 


27 


33 


40 


48 


57 


3 


11 


15 


21 


27 


33 


40 


48 


57 


1'08 


4 


20 


26 


32 


39 


46 


56 


1'07 


1'19 


137 


5 


45 


53 


1'03 


1'16 


1'31 


1'62 


2 21 


3 07 


428 


For Latitude 25°. 


Oh. 


05" 


10" 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


2 


08 


14 


19 


25 


31 


38 


46 


54 


1'05 


3 


12 


18 


24 


30 


37 


44 


53 


1'04 


118 


4 


23 


29 


35 


45 


53 


1'03 


1'16 


131 


152 


5 


49 


59 


I'lO 


1'24 


1'53 


2 07 


244 


346 


5 43 



69 



o 

< 
O 

w 






DECLmATIONS. 












For Latitude 27° 30'. 








+ 20° 


+ 15° 


+ 10° 


+ 5° 


0° 


-5° 


-10° 


-15° 


-20° 


Oh. 


08" 


13" 


18" 


24" 


30" 


36" 


44" 


52" 


1'02" 


2 


n 


16 


22 


28 


34 


41 


49 


I'OO 


110 


3 


17 


22 


28 


35 


42 


50 


I'OO 


1 11 


126 


4 


28 


35 


42 


50 


1'03 


I'll 


126 


143 


2 09 


5 


54 


1'05 


1'18 


1'34 


154 


224 


3 11 


4 38 


8 15 


FoK Latitude 30°. 


Oh. 


10" 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


l'C8" 


2 


14 


19 


25 


31 


38 


46 


54 


1'05 


118 


3 


20 


26 


32 


39 


47 


55 


1'06 


1 19 


136 


4 


.32 


39 


46 


52 


1'06 


1'19 


135 


157 


2 29 


S 


I'OO 


no 


1'24 


1'52 


2 07 


244 


3 46 


5 43 


13 06 


For Latitude 32° 30'. 


Oh. 


13" 


18" 


24" 


30" 


36" 


44" 


52" 


1'02" 


l']4" 


2 


17 


22 


28 


35 


42 


50 


I'OO 


1 11 


126 


3 


23 


29 


35 


43 


61 


I'Ol 


1 13 


128 


147 


4 


35 


43 


51 


I'Ol 


1'13 


127 


140 


213 


254 


5 


1'03 


1'15 


I'ol 


153 


2 20 


3 05 


425 


7 36 




FoK Latitude 35°. 


Oh, 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


1'08" 


121" 


2 


20 


25 


32 


38 


46 


55 


105 


118 


135 


3 


26 


33 


39 


47 


66 


1'07 


121 


138 


2 00 


4 


39 


47 


56 


1'07 


1'20 


1£6 


1 59 


2 32 


3 25 


5 


1'07 


1'20 


I'as 


2 00 


2 34 


3 29 


5 14 


10 16 




Foii Latitude 37° 30'. 


Oh. 


18" 


24" 


30" 


36" 


44" 


52" 


1'02" 


1'14" 


1'29" 


2 


22 


28 


35 


42 


50 


I'OO 


112 


126 


145 


3 


29 


36 


43 


52 


1'02 


114 


129 


149 


2 16 


4 


43 


51 


I'Ol 


1'13 


127 


149 


2 14 


2 54 


4 05 


5 


I'll 


1'26 


154 


210 


2 49 


3 55 


6 15 


14 58 




For Latitude 40°. 


Oh. 


21" 


27" 


80" 


40" 


48" 


57" 


I'OS" 


1'21" 


1'33" 


2 


25 


82 


89 


46 


52 


1'06 


1 19 


135 


157 


3 


33 


40 


48 


57 


1'03 


121 


138 


2 02 


2 36 


4 


47 


55 


1'06 


1'19 


136 


158 


2 30 


3 21 


4 59 


5 


1'15 


1'31 


la 


2 20 


3 05 


4 25 


734 


2518 




FoK Latitude 42° 30'. 


Oh 


24" 


80" 


80" 


44" 


52" 


1'02" 


1'14" 


1'29" 


]'49" 


2 


28 


85 


89 


50 


I'OO 


112 


126 


145 


211 


3 


36 


43 


52 


1'02 


113 


129 


149 


2 17 


2 59 


4 


50 


I'OO 


I'll 


126 


144 


210 


2 49 


355 


616 


5 


1'16 


186 


158 


2 80 


3 22 


5 00 


924 







60 



o 
< 

o 
a 


DECLINATIONS. 


For Latitude 45°. 


+ 20° 


+ 15' 


+ 10° 


+ 5° 


0° 


-5^ 


-10' 


-15' 


-20' 


Oh. 


27" 


33" 


40" 


48" 


57" 


I'OB" 


1'21" 


1'39" 


2'02" 


a 


32 


39 


46 


52 


l'C6 


1 19 


135 


157 


2 39 


3 


40 


47 


56 


1'07 


121 


138 


2 00 


234 


3 29 


4 


54 


1'04 


.1'16 


133 


154 


2 24 


3 11 


4 38 


815 


5 


1'23 


141 


2 05 


2 41 


3 40 


5 40 


12 03 






For Latitude 47° 30'. 


Oh. 


80" 


30" 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49" 


2'18" 


a 


35 


42 


50 


I'OO 


112 


126 


145 


2 01 


2 51 


3 


43 


51 


I'Ol 


1 13 


128 


147 


2 15 


2 56 


4 08 


4 


56 


1'03 


123 


140 


2 05 


2 40 


3 39 


5b7 


11 18 


5 


I':;? 


14Q 


2 13 


2 53 


4 01 


6 30 


1619 






Fob Latitude 50". 


Oh. 


33" 


40" 


48" 


57" 


1'08" 


I'Cl" 


1'39" 


202" 


2'36" 


2 


38 


46 


55 


l'C6 


118 


135 


1 57 


2 28 


3 19 


3 


47 


56 


ro6 


119 


136 


2 29 


2 31 


3 S3 


5 02 


4 


1'03 


l']4 


129 


148 


2 16 


2 58 


4 18 


6 59 


19 47 


5 


130 


1 51 


219 


304 


4 S3 


7 28 


34 10 






Fob Latitude 52° SO'. 


Ob. 


36" 


44" 


52" 


1'03" 


1'14" 


1'29" 


l'^9" 


2'18" 


3 '05" 


2 


43 


50 


59 


111 


126 


143 


2 23 


2 49 


3 55 


8 


50 


I'OO 


I'll 


123 


145 


211 


3 51 


2 53 


6 23 


4 


1'05 


118 


135 


2 10 


2 28 


3 19 


4 53 


842 




5 


134 


156 


2 27 


316 


4 47 


8C3 








Fob Latitude 55°. 


Oh. 


40" 


43" 


57" 


I'OS" 


1'21" 


1'39" 


2 '02" 


2'36" 


3'r3" 


2 


46 


55 


1'05 


118 


134 


156 


2 30 


315 


4 47 


3 


55 


1'06 


119 


135 


1 58 


2 30 


3 21 


4 (.3 


9 19 


4 


I'lO 


133 


142 


2 06 


2 43 


341 


5 49 


12 41 




5 


137 


2 01 


2 34 


3S8 


515 


10 13 








Foe Latitude 57° SO'. 


Oh. 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49"- 


2' 18" 


3'05" 


4'C7" 


2 


50 


59 


111 


125 


143 


2 09 


2 47 


3 51 


6C4 


3 


58 


I'lO 


124 


143 


2 07 


2 43 


3 45 


5 50 


12 47 


4 


I'll 


125 


143 


210 


2 50 


3 55 


6 14 


14 49 




5 


141 


2 06 


2 42 


342 


5 46 


12 26 








Foe Latitude 60°. 


Oh. 


48" 


57" 


1'08" 


1'21" 


1'39" 


2'02" 


3'36" 


3'33" 


5'23" 


2 


54 


1'04 


117 


133 


154 


2 24 


3 12 


4 38 


815 


3 


1'03 


115 


130 


151 


220 


3 04 


4 24 


7 31 


24 44 


4 


118 


134 


156 


2 28 


318 


4 50 


8 53 






5 


145 


3 11 


2 50 


3 57 


6 21 


15 32 









61 



AN ELEMENTARY TREATISE 



SURVEYING AND NAVIGATION 



BY 



ARTHUR G. ROBBINS, S.B. 

ASSISTANT PROFESSOR OF CIVIL ENGINEERING, MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY 




LEACH, SHEWELL, AND SANBORN 

BOSTON NEW YORK CHICAGO 



\ 



LIBRARY OF CONGRESS 



020 186 147 




;^< 






